
    UNITED STATES of America, Plaintiff-Appellee, v. Kenneth Don WILLIAMS, Defendant-Appellant. UNITED STATES of America, Plaintiff-Appellee, v. Michael John MULLINS, Defendant-Appellant.
    Nos. 92-9005, 93-1291.
    United States Court of Appeals, Fifth Circuit.
    April 20, 1994.
    
      Timothy Crooks, Asst. Federal Public Defender, Ft. Worth, TX, Paul D. Stiekney, Asst. Federal Public Defender, Ira Kirken-doll, Federal Public Defender, Dallas, TX, for Kenneth Don Williams.
    Michael Snipes, Asst. U.S. Atty., Marvin Collins, U.S. Atty., Richard H. Stephens, Dallas, TX, for U.S. in Case no. 92-9005.
    Timothy Crooks, Asst. Federal Public Defender, Fort Worth, TX, Susan Bronstein Dunleavy, Asst. Federal Public Defender, Ira Kirkendoll, Federal Public Defender, Dallas, TX, for Michael John Mullins.
    Candína S. Heath, Asst. U.S. Atty., Paul E. Coggins, Dallas, TX, for U.S. in Case no. 93-1291.
    Before GOLDBERG, .DAVIS and DeMOSS, Circuit Judges.
   GOLDBERG, Circuit Judge:

We consider in these appeals the appellants’s challenges to the trial court’s definition of reasonable doubt. Guided by the Supreme Court’s recent discussion of this issue in Victor v. Nebraska, — U.S. -, 114 S.Ct. 1239, 127 L.Ed.2d 583 (1994), and by the realization that no court can guarantee the absolute certitude of any definition of reasonable doubt, we find the instructions given' by the district court to be acceptable. We also find that the appellants’s individual contentions do not warrant reversal of their convictions. We therefore affirm the judgments of the district court.

I. Facts and Proceedings Below

Kenneth Don Williams was' charged in a one count indictment of being a convicted felon in possession of a firearm in violation of 18 U.S.C. §§ 922(g)(1), 924(a)(2), and 924(e). In a separate and unrelated indictment, Michael John Mullins was charged "with two counts of being a convicted felon in possession of a firearm in violation of 18 U.S.C. §§ 922(g)(1) and 924(e)(1). Both men pleaded not guilty, and each proceeded to trial.

At both trials, the juries were instructed that the government had to prove each element of the charged offenses beyond a reasonable doubt. The district court gave the juries the same definition of reasonable doubt in both cases. That definition reads as follows:

Proof beyond a reasonable doubt is proof that leaves you firmly convinced of a defendant’s guilt. There are few things in life that we know with absolute certainty, and in criminal eases the law does not require proof that a defendant is guilty beyond all possible doubt. If, based on your consideration of all the evidence, you are firmly convinced that a defendant is guilty of the crime charged, you must find him guilty. If, however, you think there is a real possibility that he is not guilty, you must give him the benefit of the doubt and find him not guilty.

(emphasis added by the appellants).

The jury in Williams’s case convicted him on the one count on which he was charged. He was sentenced to serve a 240 month term of imprisonment. The jury in Mullins’s ease convicted him on the first felon in possession of a firearm count on which he was charged, but could not reach a verdict on the second count. After his trial but before sentencing, Mullins filed a motion to have four prior Texas state felony convictions against him invalidated for the purposes of sentencing. The district court denied this motion and used three of these prior convictions to enhance Mullins’s sentence under the Armed Career Criminal Act, 18 U.S.C. § 924(e). Mullins was sentenced to 235 months of imprisonment.

Both defendants appeal, contending that the district court’s definition of reasonable doubt understated the level of proof that the government must meet to win a conviction and overstated the level of uncertainty necessary before the jury must acquit a defendant. Each defendant also raises matters specific to his own conviction. We will examine these issues in turn.

II. Discussion

A. Reasonable Doubt

The Due Process Clause of the Constitution requires the government to prove every element of a charged offense beyond a reasonable doubt. In re Winship, 397 U.S. 358, 90 S.Ct. 1068, 25 L.Ed.2d 368 (1970). The Supreme Court has recently observed that, “[although this standard is an ancient and honored aspect of our criminal justice system, it defies easy explication.” Victor v. Nebraska, — U.S. at -, 114 S.Ct. at 1242. Perhaps for this reason, neither the Supreme Court nor this Court have ever required a particular definition of reasonable doubt to be read to the jury. See id. at -, 114 S.Ct. at 1243 (“[S]o long as the court instructs the jury on the necessity that the defendant’s guilt be proven beyond a reasonable doubt, the Constitution does not require that any particular form of words be used in advising the jury of the government’s burden of proof.”) (citation omitted). Nevertheless, any definition of reasonable doubt that a district court does use must, “taken as a whole, ... correctly convey[ ] the concept of reasonable doubt to the jury.” Holland v. United States, 348 U.S. 121, 140, 75 S.Ct. 127, 138, 99 L.Ed. 150 (1954). If there is a “reasonable likelihood that the jury understood the instructions to allow conviction based on proof insufficient to meet the Win-ship standard,” then the instruction runs afoul of the Due Process Clause, and the conviction must be reversed. Victor, — U.S. at -, 114 S.Ct. at 1243.

In this case, the Constitution is not our only benchmark. Through our supervisory powers, we “may, within limits, formulate procedural rules not specifically required by the Constitution or the Congress.” United States v. Hasting, 461 U.S. 499, 505, 103 S.Ct. 1974, 1978, 76 L.Ed.2d 96 (1983). Thus, in order “to preserve judicial integrity,” id., we may reverse the appellant’s convictions if we are persuaded that the district court’s jury instruction is not acceptable, even though it passes constitutional muster. In these cases, however, we find that under any standard, the instruction that the district judge gave acceptably defined reasonable doubt.

The definition of reasonable doubt that the district court gave the juries in the present cases is similar to a definition of reasonable doubt endorsed by the Federal Judicial Center. See Federal Judicial Center, Pattern Criminal Jury Instructions 17-18 (1987) (instruction 21). However, the parentage of the district court’s definition of reasonable doubt is not all that recommends it; previous panels of this Court have explicitly endorsed the very instruction that the district court used in these cases. See United States v. Hunt, 794 F.2d 1095 (5th Cir.1986); United States v. Haggard, 5 F.3d 1494 (5th Cir.1993) (unpublished).

Nevertheless, the appellants argue that the district court’s definition of reasonable doubt, and by implication this Court’s opinion in Hunt and cases that have followed it, were drawn into question by Cage v. Louisiana, 498 U.S. 39, 111 S.Ct. 328, 112 L.Ed.2d 339 (1990) (per curiam). In that case, a Louisiana state court trial judge instructed the jurors as follows:

[A reasonable doubt] is one that is founded upon a real tangible substantial basis and not upon mere caprice and conjecture. It must be such doubt as would give rise to a grave uncertainty, raised in your mind by reasons of the unsatisfactory character of the evidence or lack thereof. A reasonable doubt is not a mere possible doubt. It is an actual substantial doubt. It is a doubt that a reasonable man can seriously entertain. What is required is not an absolute or mathematical certainty, but a moral certainty.

Id. at 40, 111 S.Ct. at 329 (emphasis supplied by the Court). The emphasized portions of the instruction in Cage rendered the charge used in that case unconstitutional because it “suggest[ed] a higher degree of doubt than is required for acquittal under the reasonable doubt standard.” Id. at 41, 111 S.Ct. at 329-330. The Court explained that when the highlighted portions of the charge were then “considered with the reference to ‘moral certainty,’ rather than evidentiary certainty, it becomes clear that a reasonable juror could have interpreted the instruction to allow a finding of guilt based on a degree of proof below that required by the Due Process Clause.” Id.

After this case was argued, the Supreme Court handed down its opinion in Victor v. Nebraska, — U.S. -, 114 S.Ct. 1239. In that case, the Court considered and rejected constitutional challenges to two other state court definitions of reasonable doubt. In the first case, a California state court defined reasonable doubt as follows:

[Reasonable doubt] is not a mere possible doubt; because everything relating to human affairs, and depending on moral evidence, is open to some possible or imaginary doubt. It is that state of the case which, after the entire comparison and consideration of all the evidence, leaves the minds of the jurors in that- condition that they cannot say they feel an abiding conviction, to a moral certainty, of the truth, of the charge.

Id. at -, 114 S.Ct. at 1244. The Court rejected the defendant’s contention that use of the terms “moral evidence” and “moral certainty” rendered the charge unconstitutional. The Court ruled that the reference to “moral evidence” did not focus the jury’s attention on the ethics or morality of the defendant’s acts; instead, the Court held that the charge, taken as a whole, adequately instructed the jury to consider the facts of the case. Id. at -, 114 S.Ct. at 1247. Similarly, the Court held that the reference to “moral certainty”, although more problematic, and ambiguous in the abstract, did not render the instruction given in that case unconstitutional. Other parts of the charge (particularly the “abiding conviction” language) supplied the instruction with the necessary content. The Court concluded that the charge “ ‘impress[ed] upon the factfinder the need to reach a subjective state of near certitude of the guilt of the accused.’ ” Id. (quoting Jackson v. Virginia, 443 U.S. 307, 315, 99 S.Ct. 2781, 2787, 61 L.Ed.2d 560 (1979)). Finally, the Court held that the charge’s instruction that a reasonable doubt was “not a mere possible doubt” was not objectionable. This was for the simple reason that ‘[a] “reasonable doubt,” at a minimum, is one based upon a “reason.”’ A fanciful doubt is not a reasonable doubt.” Id. at -, 114 S.Ct. at 1248 (quoting Jackson, 443 U.S. at 317, 99 S.Ct. at 2788).

In the second case that the Supreme Court considered in Victor, a Nebraska state court defined reasonable doubt as follows:

“Reasonable doubt” is such a doubt as would cause a reasonable and prudent person, in one of the graver and more important transactions of life, to pause and hesitate before taking the represented facts as true and relying and acting thereon. It is such a doubt as will not permit you, after full, fair, and impartial consideration of all the evidence, to have an abiding conviction, to a moral certainty, of the guilt of the accused. At the same time, absolute or mathematical certainty is not required. You may be convinced of the truth of a fact beyond a reasonable doubt and yet be fully aware that, possibly you may be mistaken. You may find an accused guilty upon the strong probabilities of the case, provided such probabilities are strong enough to exclude any doubt of his guilt that is reasonable. A reasonable doubt is an actual and substantial doubt arising from the evidence, from the facts or circumstances shown by the evidence, or from the lack of evidence on the. part of the state, as distinguished from a doubt arising from mere possibility, from bare imagination, or from fanciful conjecture.

Id. at -, 114 S.Ct. at 1249. The Court acknowledged that defining reasonable doubt to be actual and substantial doubt was somewhat problematic because such a definition could be read to overstate the degree of doubt required for acquittal. Indeed, in Cage, the use of similar language led the Court to hold that the definition of reasonable doubt given in .that case violated the Due Process Clause. Cage, 498 U.S. at 41, 111 S.Ct. at 329-330. However, the context in which the suspect language appeared sufficiently eviscerated the difficulty with this feature of the charge. The instruction itself explained that “an actual and substantial doubt” was to be “distinguished from a doubt arising from mere possibility, from bare imagination, or from fanciful conjecture.” Victor, — U.S. at -, 114 S.Ct. at 1249. With this admonition, the Court held, the charge correctly instructed the jury that actual and substantial doubt was doubt that was not seeming or imaginary. Id. at -, 114 S.Ct. at 1250. The Court also rebuffed the defendant’s challenge to the “moral certainty” language contained in the charge used at his trial. As the Court wrote, “[i]n-strueting the jurors that they must have an abiding conviction of the defendant’s guilt does much to alleviate any concerns that the phrase moral certainty might be misunderstood in the abstract.” Id. The Court also observed that the instruction “equated a doubt sufficient to preclude moral certainty with a doubt that would cause a reasonable person to hesitate to act,” id. at — - —, 114 S.Ct. at 1250-1251, a definition of reasonable doubt that the Court approved of in Holland. 348 U.S. at 140, 75 S.Ct. at 137-138. Finally, the Court rejected the defendant’s argument that the charge’s reference to “strong probabilities” understated the government’s burden of proof. The same sentence of the charge informed the jury that “the probabilities must be strong enough to prove the defendant’s guilt beyond a reasonable doubt.” Victor, — U.S. at —, 114 S.Ct. at 1251.

In the present cases, Williams and Mullins challenge the “firmly convinced” language that appears in the district court’s charge. They contend that the charge understated the government’s burden of proof by describing it to be closer to the preponderance of the evidence standard than the constitutionally required beyond a reasonable doubt standard. We are not persuaded. The appellants cannot complain that the “firmly convinced” formulation speaks in terms of probabilities because “the beyond a reasonable doubt standard is itself probabilistic.” Id. at -, 114 S.Ct. at 1247. The appellants can only complain that the phrase “firmly convinced” connotes something less than the “very high level of probability required by the Constitution in criminal cases.” Id. However, we think that the “firmly convinced” language, read in the context of the charge as a whole, adequately apprises the jury of the requisite level of proof. By repeating the admonition that the jury had to be firmly convinced of the appellants’s guilt and by explaining that the government did not have to prove the appellants guilty beyond all possible doubt, the district judge sufficiently communicated to the jury that they had to find the appellants guilty to a near certainty.

The appellants also challenge the district court’s characterization of a reasonable doubt as a “real possibility” that the defendant is not guilty. We find no infirmity in this portion of the charge either. When read in the context of the charge as a whole, the instruction’s “real possibility” formulation explains that the beyond a reasonable doubt standard does not require “proof that overcomes every possible doubt.” In other words, the modifier “real” merely indicates that the jury is not to acquit a defendant if it can conceive of any possibility that the defendant is not guilty. This is because “absolute certainty is unattainable in matters relating to human affairs.” Id. at -, 114 S.Ct. at 1246. Just as the jury is not to indulge in fanciful speculation that the defendant is guilty, the jury is not to indulge in fanciful speculation that the defendant is not guilty.

A definition of reasonable doubt that includes the “firmly convinced” and “real possibility” language was expressly endorsed and ably defended by Justice Ginsburg in her concurrence in Victor. — U.S. at -, 114 S.Ct. at 1252 (Ginsburg, J., concurring in part and concurring in the judgment). After setting out the Federal Judicial Center’s proposed definition of reasonable doubt (an instruction nearly identical to the one given by the district court in these cases), see supra note 2, Justice Ginsburg explained that “[t]he ‘firmly convinced’ standard for conviction, repeated for emphasis, is ... enhanced by the juxtaposed prescription' that the jury must acquit if there is a ‘real possibility’ that the defendant is innocent.” Victor, — U.S. at -, 114 S.Ct. at 1253. This recommendation further supports our own approval of the district court’s instruction.

In short, nothing in Cage or Victor persuades us that the trial court’s definition of reasonable doubt is unacceptable. Composing a perfect definition of reasonable doubt may be an illusory goal, but perfection and certitude are rare in any intellectual discipline, whether it be scientific or, humanistic. There are very few things that can .be said with an assurance that is unimpeachable. Since Einstein’s day, we have been told that even in the sciences, conclusions do not work out that way; there is almost always a predicate of fluidity and relativity. With this understanding, we have examined the boundaries of the words contained in the district court’s charge and the longitude and latitude of their expressiveness. This examination leaves us with the firm conviction that the jury, upon hearing the judge’s instructions, knew how to determine whether there was a reasonable doubt as to the defendants’s guilt.

Finally, the appellants assert that the instruction that the district court gave impermissibly deviated from the Fifth Circuit’s Pattern Jury Instructions. We find no merit to this argument. Although the Pattern Jury Instructions provide a useful guide for the district courts, we have never required the trial courts in this Circuit to use any particular language in a jury charge. See United States v. Masat, 948 F.2d 923, 928 (5th Cir.1991) (“Trial judges have substantial latitude in tailoring their instructions if they fairly and adequately cover the issues presented in the case.”), cert. denied, — U.S. -, 113 S.Ct. 108, 121 L.Ed.2d 66 (1992).

B. Mullins’s Individual Contentions

Mullins contends that the district court erroneously refused to invalidate for the purpose of sentencing four state felony convictions to which he had pleaded guilty on November 24,1986. Three of these four convictions were for violations of the Texas Controlled Substances Act; the fourth conviction was for theft. The violations of the Controlled Substances Act were “serious drug offenses” under the Armed Career Criminal Act (“ACCA”), 18 U.S.C. § 924(e), and were thus used to enhance Mullins’s sentence under that law. Mullins maintains that these convictions were constitutionally invalid because his guilty pleas were not voluntarily and intelligently given. See North Carolina v. Alford, 400 U.S. 25, 91 S.Ct. 160, 27 L.Ed.2d 162 (1970). Specifically, he argues that he never actually declared that he was guilty of the four offenses, that he was not adequately advised of the nature of the charges against him, and that a sufficient factual basis for his guilty pleas did not exist.

The district court held a hearing concerning the constitutional validity of Mullins’s 1986 convictions. At this hearing, the district court examined a copy of the transcript of the November 24, 1986 state court arraignment and sentencing hearing. The district court determined that Mullins had knowingly and intelligently pleaded guilty to the four challenged state court convictions. Agreeing with the district court below, we conclude that Mullins’s state court guilty pleas were properly accepted. The record reflects that Mullins voluntarily and intelligently pleaded guilty and had an understanding of the rights that he waived and the consequences of his pleas.

Mullins’s first argument is that since he never actually declared that he was guilty of the four offenses, he never entered a valid plea of guilty to the four crimes. However, an examination of the record belies Mullins’s contention that he did not plead guilty. In the state court proceeding, the court first informed Mullins of the nature and elements of the offenses with which he was charged and determined that he understood the charges against him. The court then questioned Mullins as follows:

THE COURT: Papers that are filed in your eases indicate to the Court that when you are arraigned you will plead guilty, is that true?
MULLINS: Yes, sir.
THE COURT: Are you pleading guilty of your own free will and accord?
MULLINS: Yes, sir.
THE COURT: Has anyone promised you anything, coerced you or threatened you or done any violence to you to make you plead guilty in any of these four cases?
MULLINS: No, sir.

The court then informed Mullins of the range of punishment that he was facing for each crime. Next, Mullins’s own attorney questioned him:

Q: Michael, you are the same person named in these four indictments, is that correct?
A. Right.
Q: Did I explain to you your right to have separate jury trials in these cases?
A: Right.
Q: I had your permission to give up those rights?
A: Yes, sir.
Q: And in entering you [sic] plea of guilty, have I had sufficient time to go over the facts of the cases with you?
A: Yes, sir.

The court then admitted a signed Judicial Confession for each crime. In these confessions, Mullins waived his constitutional rights and confessed to committing the crimes alleged in the four indictments. Later, the prosecutor questioned Mullins:

Q: You signed a judicial confession in each of the four cases?
A. Yes, I guess I did, I did, yes.
Q: Did you commit the offenses as those confessions say you did?
A: Yes.

These colloquies, when coupled with the written confession that Mullins signed for each offense, demonstrate that the defendant did in fact plead guilty to the four charges against him. In the context of this case, it is of no consequence that Mullins did-not specifically utter the words “I am guilty.” Although this is plainly the better course, we do not require such a talismanic incantation, so long as the language used.is expressive of the defendant’s culpability.

Furthermore, a review of the state court arraignment and sentencing hearing also reveals that Mullins’s remaining contentions cannot withstand scrutiny. The record reveals that Mullins was adequately advised of the nature of the charges against him and that there was a sufficient factual basis for his guilty pleas.

Finally, at his federal sentencing hearing, Mullins claimed to be high on heroin when he entered his 1986 guilty pleas. However, at the federal hearing, Mullins acknowledged that, during his state hearing, he understood that he was pleading guilty to four separate counts, that he signed separate confessions for each offense, and that he committed each of the offenses with which he was charged. Mullins also acknowledged that he was telling the truth at the state sentencing hearing. Mullins has failed to show that any drugs that he may have been taking so affected him that he was incapable of making a voluntary and intelligent waiver of his trial rights. Cf. Godinez v. Moran, — U.S. —, 113 S.Ct. 2680, 125 L.Ed.2d 321 (1993) (test of mental competency to plead guilty is whether the defendant has a sufficient present ability to consult with a lawyer with a reasonable degree of rational understanding and has a rational as well as factual understanding of the proceedings against him.).

In sum, we conclude that the district court did not' err when it concluded that Mullins had knowingly and intelligently waived his rights and validly pleaded guilty to the charges against him. '

C. Williams’s Individual Contentions

Williams argues that the government committed prosecutorial misconduct to a degree sufficient to warrant reversal of his conviction. To rise to this level, prosecutorial misconduct must be “so pronounced and persistent that it casts serious doubts upon the correctness of the jury’s verdict.” United States v. Bentley-Smith, 2 F.3d 1368, 1378 (5th Cir.1993). The misconduct that Williams alleges takes two forms. Williams first contends that the prosecutor elicited improper character evidence against him and improperly impeached Shazelle Williams, the defendant’s wife. Williams also' objects to allegedly improper remarks that the prosecutor made in his closing argument. Only some of these objections were made during the trial. After a studied review the record, we hold these instances of alleged misconduct do not require reversal of Williams’s conviction.

III. Conclusion

The judgments of the district court are AFFIRMED. 
      
      . Believing that all definitions of reasonable doubt are damaging, at least two Federal Courts of Appeals have advised against making any such attempt. See United States v. Adkins, 937 F.2d 947, 950 (4th Cir.1991) ("This circuit has repeatedly warned against giving the jury definitions of reasonable doubt, because definitions tend to impermissibly lessen the burden of proof."); United States v. Hall, 854 F.2d 1036, 1039 (7th Cir.1988) (“[N]o attempt should be made to define reasonable doubt.... [T]he point is that, at best, definitions of reasonable doubt are unhelpful to a jury, and, at worst, they have the potential to impair a defendant’s constitutional right to have the government prove each element beyond a reasonable doubt. An attempt to define reasonable doubt presents a risk without any real benefit.”). In contrast, we have encouraged the district courts in this Circuit to use this Circuit’s Pattern Jury Instruction on the definition of reasonable doubt. See infra note 2.
     
      
      . The Federal Judicial Center's proposed definition of reasonable doubt reads as follows:
      [T]he government has the burden of proving the defendant guilty beyond a reasonable doubt. Some of you may have served as jurors in civil cases, where you were told that it is only necessary to prove that a fact is more likely true than not true. In criminal cases, the government’s proof must be more powerful than that. It must be beyond a reasonable, doubt.
      Proof beyond a reasonable doubt is proof that leaves you firmly convinced of the defendant's guilt. There are very few things in this world that we know with absolute certainty, and in criminal cases the law does not require' proof that overcomes every possible doubt. If, based on your consideration of the evidence, you are firmly convinced that the defendant is guilty of the crime charged, you must find him guilty. If on the other hand, you think there is a real possibility that he is not guilty, you must give him the benefit of the doubt and find him not guilty.
      Federal Judicial Center, Pattern Criminal Jury Instructions 17-18 (1987) (instruction 21).
      The Fifth Circuit's Pattern Jury Instructions for criminal cases offers an alternative definition of reasonable doubt. It defines the government’s burden of proof in a criminal case as follows:
      A "reasonable doubt” is a doubt based upon reason and common sense after careful and impartial consideration of all the evidence in the case. Proof beyond a reasonable doubt, therefore, is proof of such a convincing character that you would be willing to rely and act upon it without hesitation in the most important of your own affairs.
      United States Fifth Circuit District Judges Association, Pattern Jury Instructions (Criminal Cases) 16 (1990) (instruction 1.06). Although we do not require the use of this instruction, we have encouraged the district courts in this Circuit to adopt this instruction, hoping that “[a] measure of uniformity would ... render appellate review easier and quicker.” United States v. Hunt, 794 F.2d 1095, 1101 (5th Cir.1986).
     
      
      . In Victor, the Court was careful to note that it was not the Cage instruction's reference to substantial doubt alone that rendered the charge in that case unconstitutional. Instead, the Cage Court was "concerned that the jury would interpret the term 'substantial doubt' in parallel with the preceding reference to 'grave uncertainty,' leading to an overstatement of the doubt necessary to acquit.” Victor, — U.S. at -, 114 S.Ct. at 1250.
     
      
      . The appellants first contend that the district court's use of the term “firmly convinced” is directly analogous to the use of the term "moral certainty” in Cage. We are somewhat perplexed by this argument. To be sure, in Cage and Victor, the Court expressed concern with the use of the phrase “moral certainty”. But the Court's complaint with that term stemmed from its fear that the expression might not be recognized by modem juries. Victor, — U.S. at -, 114 S.Ct. at 1247. No such problem is present with the "firmly convinced” language that the district court used in the present cases. Instead, the appellants’s best argument is that the term “firmly convinced” understates the level of proof that the government must meet. We address this argument in the text.
     
      
      . We think that the Federal Judicial Center's instruction is superior to the instruction that the district court gave in at least one respect. The Federal Judicial Center’s instruction informs the jurors that the prosecution must prove its case by more than a mere preponderance of the evidence, but not necessarily to an absolute certainty. The district court's instruction does not contrast the preponderance of the evidence standard and the beyond a reasonable doubt standard. Such a contrast is a useful way to frame the issue for the jury.
     
      
      . Mullins has two other felony convictions — a 1974 conviction for delivery of marijuana- and a 1976 conviction for the sale of marijuana- — that he does not challenge.
      As noted below, a person who has been convicted under 18 U.S.C. § 922(g) and has three or more prior "serious drug offenses," is subject to an enhanced sentence under 18 U.S.C. § 924(e). All five of Mullins's prior drug convictions meet the statutory definition of a "serious drug offense”; thus, all five of these convictions were counted towards the application of § 924(e). If Mullins's attack on the four 1986 felony convictions has merit (particularly the three counts for violating the Texas Controlled Substances Act), then he would have only two "serious drug offenses,” and § 924(e) would not be applicable.
     
      
      .Section 924(e) provides:
      In the case of a person who violates section 922(g) of this title and has three previous convictions by any court referred to in section 922(g)(1) of this title for a violent felony or a serious drug offense, or both, committed on occasions different from one another, such person shall be fined not more than $25,000 and imprisoned not less than fifteen years, and ... such person shall not be eligible for parole with respect to the sentence imposed under this subsection. 18 -U.S.C. § 924(e)(1).
      Furthermore, although Mullins's initial offense level was 24, because Mullins was subject to an enhanced sentence as an armed career criminal under § 924(e), his adjusted offense level was required to be'33. See U.S.S.G. § 4B1.4(b).
     
      
      . In such a hearing, the government has the initial burden of submitting evidence of past convictions that exposes the defendant to punishment under 18 U.S.C. § 924(e). Unless such evidence reveals unconstitutionality on its face, it will suffice to prove the existence of valid convictions. Once the government establishes the fact of the prior convictions, the defendant must then prove the constitutional invalidity of apy conviction that he or she attacks by a preponderance of the evidence. United States v. Barlow, 17 F.3d 85, 89 (5th Cir.1994). The district court’s factual findings will be upheld unless they are clearly erroneous. Id.
      
     
      
      . This argument is premised on the proposition that, in order for there to be a valid plea of guilty, a defendant must personally utter the word "Guilty” after the court asks "How do you plead?” We have found no cases that contain such a holding. Instead, our review of the case law demonstrates that "there is no fixed colloquy, no set sequence or number of questions and answers, no minimum length of the hearing, and no talismanic language” that is required to be used in guilty-plea hearings. Stewart v. Peters, 958 F.2d 1379, 1384 (7th Cir.), cert. denied, - U.S. -, 113 S.Ct. 239, 121 L.Ed.2d 173 (1992).
     