Jerome Damasco filed this putative class-action lawsuit against Clearwire Corporation in an Illinois state court, alleging that Clearwire violated the Telephone Consumer Protection Act, 47 U.S.C. § 227, by sending unsolicited text messages to cellphone users. Before Damasco moved for class certification, Clearwire offered him his full request for relief. Clearwire then removed the case to federal court and moved to dismiss, arguing that the offer mooted Damasco’s claim. The district court agreed, dismissed Damasco’s complaint, and later denied his motion to reconsider. Damasco appeals both rulings. Under Holstein v. City of Chicago, 29 F.3d 1145, 1147 (7th Cir.1994), Clearwire’s offer mooted Damasco’s claim. We thus affirm the court’s judgment and its decision to deny reconsideration.

I.

Damasco asked the state court to enjoin Clearwire from sending unsolicited text messages and to grant damages to all those injured by this practice. See 47 U.S.C. § 227(b)(3). He estimated that more than 1,000 people had received these messages and requested damages fixed by the Act, $500 for each violation. See id. § 227(b)(3)(B). Damasco added that the court could award three times that amount, up to $1,500 for each violation, if it determined that Clearwire had acted “willfully and knowingly.” See id. § 227(b)(3)(C).

Within a month, Clearwire sent a letter to Damasco’s attorneys offering to settle the case by giving Damasco and up to ten other affected people $1,500 for each text message received from Clearwire, plus court costs. In addition, Clearwire offered to stop sending unsolicited text messages to “mobile subscribers.” Clearwire warned that, in its view, this offer rendered the case moot. Damasco never responded to Clearwire’s letter.

Four days after sending the letter, Clearwire removed the suit to federal court. Damasco moved for class certification within a few hours of the removal. The following day, Clearwire moved to dismiss the case, arguing that its settlement offer stripped Damasco of his personal stake in the case’s outcome and rendered his claim moot.

Damasco opposed Clearwire’s motion. He contended that Clearwire’s letter did not constitute an offer under Illinois law because its terms were not “definite and certain.” But even if the offer was valid, he urged that the controversy remained live, primarily for three reasons. First, he insisted that defendants should be prohibited from mooting a potential class action by buying off named plaintiffs through “involuntary” settlements. Second, he argued that this type of claim is “inherently transitory” — that is, bound to become moot before the class is certified — so his motion for certification should “relate back” to the filing of his complaint, as permitted in Sosna v. Iowa, 419 U.S. 393, 402 n. 11, 95 S.Ct. 553, 42 L.Ed.2d 532[*894] (1975). Finally, he maintained that if Clearwire had made an offer under Federal Rule of Civil Procedure 68, then he would have had 10 days (now 14 days under a revised version of the rule) to ask the court to certify the class and avoid mootness. He argued that Clearwire should not be allowed to circumvent Rule 68 by casting its offer in the form of a settlement.

The district court agreed with Clearwire and dismissed the case. Finding the settlement offer to be sufficiently definite under Illinois law, the court ruled the case moot. The court observed, citing Holstein, that “[t]he rule in the Seventh Circuit is clear- — -a complete offer of settlement made prior to the filing for class certification moots the plaintiffs claim.” The court acknowledged Damasco’s concerns about defendants buying off class representatives, but emphasized that “Holstein has not been overturned and it is directly on point.” Some district courts, the court noted, have allowed plaintiffs to avoid mootness by seeking class certification after being offered complete relief under Rule 68. But when, as here, an offer is “not made pursuant to Rule 68,” the court reasoned, applying that rule’s timeframe for accepting an offer would be “arbitrary.” The court also rejected Damasco’s argument that his claim was inherently transitory.

Damasco moved for reconsideration, arguing that “new evidence” showed that Clearwire’s offer was not valid. He pointed to Fahey v. Career Education Corp., No. 1:10-ev-05635 (N.D.Ill.) — a similar lawsuit pending before the same district judge with the same attorneys but different parties — -where defense counsel sent an offer to plaintiffs attorney that was nearly identical to the offer here. Damasco claimed that the named plaintiff in Fa-hey believed that she had accepted the offer, but that the defendant was proceeding as if no agreement had been reached.

Before ruling on the motion for reconsideration, the district court held a hearing in Fahey to determine whether a settlement had been reached. After the court concluded that no agreement had been reached, the plaintiffs attorney immediately asked the court whether the offer mooted the case. The court replied that it did not. But when pressed for clarification by defendant’s counsel, the court revised its comments, explaining that the hearing had not dealt with mootness, only whether the case was settled. Shortly after this hearing, Damasco supplemented his motion for reconsideration, arguing that “[i]f the same letter did not moot Fahey’s claim, then it cannot have mooted Damasco’s claim.”

The district court then denied Damasco’s motion for reconsideration. The court remarked that “even if all the alleged conduct from Fahey happened in this case, the conduct amounts to dishonor of an agreement,” not “newly discovered evidence” under Federal Rule of Civil Procedure 59(e). The court explained that its comments about mootness in Fahey were “merely dicta” since “no party in that case had moved for dismissal on the basis of mootness.” The court noted that the Fa-hey defendants had since moved to dismiss and that “a full discussion of the issue of mootness in that case is better left to that context.” Fahey is currently being held in abeyance pending the resolution of this appeal.

II.

The doctrine of mootness stems from Article III of the Constitution, which limits the jurisdiction of federal courts to live cases or controversies. Spencer v. Kemna, 523 U.S. 1, 7, 118 S.Ct. 978, 140 L.Ed.2d 43 (1998); AM. v. Butler, 360 F.3d 787, 790 (7th Cir.2004). The doctrine[*895] demands that the parties to a federal case maintain a personal stake in the outcome at all stages of the litigation. United States v. Juvenile Male, — U.S. -, 131 S.Ct. 2860, 2864, 180 L.Ed.2d 811 (2011); Spencer, 523 U.S. at 7, 118 S.Ct. 978. Therefore, “[o]nee the defendant offers to satisfy the plaintiffs entire demand, there is no dispute over which to litigate, and a plaintiff who refuses to acknowledge this loses outright, under Fed.R.Civ.P. 12(b)(1), because he has no remaining stake.” Rand v. Monsanto Co., 926 F.2d 596, 598 (7th Cir.1991) (citation omitted); accord Breneisen v. Motorola, Inc., 656 F.3d 701, 706 (7th Cir.2011).

Damasco asks us to create an exception to mootness in potential class actions where defendants offer relief to named plaintiffs before they have “a reasonable opportunity to seek certification.” He points out that mootness is a “flexible” doctrine, see U.S. Parole Comm’n v. Geraghty, 445 U.S. 388, 400, 100 S.Ct. 1202, 63 L.Ed.2d 479 (1980), and argues that Holstein, which conflicts with his proposed exception, should be restricted to its facts or overturned.

Damasco starts by highlighting that the Supreme Court and this court have emphasized the importance of preventing individual buy-offs from mooting class actions. For example, the Supreme Court has held that defendants cannot prevent an appeal from a denial of certification simply by offering relief to a named plaintiff. Deposit Guar. Nat’l Bank, Jackson, Miss. v. Roper, 445 U.S. 326, 339, 100 S.Ct. 1166, 63 L.Ed.2d 427 (1980). The Court reasoned that the alternative — requiring numerous plaintiffs to file separate actions in order to prevent them from being picked off before appellate review of certification — “would frustrate the objectives of class actions” and “invite waste of judicial resources by stimulating successive suits brought by others claiming aggrievement.” Id. Along the same lines, we have long held that a defendant cannot moot a case by making an offer after a plaintiff moves to certify a class, observing that “[ojtherwise the defendant could delay the action indefinitely by paying off each class representative in succession.” Primax Recoveries, Inc. v. Sevilla, 324 F.3d 544, 546-47 (7th Cir.2003); see Greisz v. Household Bank (Ill.), N.A., 176 F.3d 1012, 1015 (7th Cir.1999); Susman v. Lincoln Am. Corp., 587 F.2d 866, 869 (7th Cir.1978).

In light of these concerns, Damasco argues that Holstein should be overruled or distinguished so as not to control the outcome in this case. In Holstein, the plaintiff filed a putative class action after the City of Chicago towed his car, arguing that the city’s towing procedures were unconstitutional. 29 F.3d at 1147. Before he moved to certify, the city offered him full reimbursement. Id. We refused to let him “spurn this offer” and avoid mootness when he “did not even move for class certification prior to the evaporation of his personal stake.” Id. The plaintiffs lack of a personal stake, we held, stripped us of jurisdiction over his claim. Id. We repeated this holding in Greisz, remarking that an offer to a named plaintiff does not moot a class action unless it “comes before class certification is sought.” 176 F.3d at 1015. We later confirmed that a plaintiff cannot avoid mootness by moving for class certification after receiving an offer of full relief. Gates v. City of Chicago, 623 F.3d 389, 413 (7th Cir.2010).

Four circuits disagree with this approach, but we have not been moved to reverse course. Those circuits, citing the flexible nature of the mootness doctrine and concerns about buy-offs, have fashioned a new rule that, absent undue delay, a plaintiff may move to certify a class and avoid mootness even after being offered[*896] complete relief. See Pitts v. Terrible Herbst, Inc., 653 F.3d 1081, 1091-92 (9th Cir.2011); Lucero v. Bureau of Collection Recovery, Inc., 639 F.3d 1239, 1249-50 (10th Cir.2011); Sandoz v. Cingular Wireless LLC, 553 F.3d 913, 920-21 (5th Cir.2008); Weiss v. Regal Collections, 385 F.3d 337, 348 (3d Cir.2004). Although these decisions address offers that, unlike Clearwire’s, were made under Rule 68, their same analysis seems to apply to any offer of complete relief. We have acknowledged that this alternative approach creates a potential exception to mootness in class actions, Wrightsell v. Cook Cnty., Ill., 599 F.3d 781, 783 (7th Cir.2010), but we have yet to adopt it and decline to do so here.

We believe that the exception created by these circuits is unnecessary. To allow a case, not certified as a class action and with no motion for class certification even pending, to continue in federal court when the sole plaintiff no longer maintains a personal stake defies the limits on federal jurisdiction expressed in Article III. See Juvenile Male, 131 S.Ct. at 2864; Lewis v. Cont’l Bank Corp., 494 U.S. 472, 477-78, 110 S.Ct. 1249, 108 L.Ed.2d 400 (1990); Holstein, 29 F.3d at 1147. That the complaint identifies the suit as a class action is not enough by itself to keep the case in federal court. Even when a “complaint clearly and in great detail describes the suit as a class action suit,” if the plaintiff does not seek class certification, then “dismissal of the plaintiffs claim terminates the suit.” Turek v. General Mills, Inc., 662 F.3d 423, 424 (7th Cir.2011); see Bd. of Sch. Comm’rs of City of Indianapolis v. Jacobs, 420 U.S. 128, 129-30, 95 S.Ct. 848, 43 L.Ed.2d 74 (1975). After Clearwire made its offer, Damasco’s federal case was over. See Greisz, 176 F.3d at 1015. (Incidentally, the case would be over even if it had remained in state court; the Illinois Supreme Court recently reaffirmed that its approach is the same as ours. See Barber v. Am. Airlines, Inc., 241 Ill.2d 450, 457-58, 350 Ill.Dec. 535, 948 N.E.2d 1042 (2011).)

A simple solution to the buy-off problem that Damasco identifies is available, and it does not require us to forge a new rule that runs afoul of Article III: Class-action plaintiffs can move to certify the class at the same time that they file their complaint. The pendency of that motion protects a putative class from attempts to buy off the named plaintiffs. See Primax, 324 F.3d at 546-47. Damasco argues that this solution would provoke plaintiffs to move for certification prematurely, before they have fully developed or discovered the facts necessary to obtain certification. See 5 Moore’s Federal Practice § 23.64[l][b], at 350 (3d ed.2011). But this objection is unpersuasive. If the parties have yet to fully develop the facts needed for certification, then they can also ask the district court to delay its ruling to provide time for additional discovery or investigation. In a variety of other contexts, we have allowed plaintiffs to request stays after filing suit in order to allow them to complete essential activities. See Fed.R.Civ.P. 56(d) (allowing stays to complete discovery before summary judgment); Newell v. Hanks, 283 F.3d 827, 834 (7th Cir.2002) (allowing stays in habeas petitions to permit exhaustion without risk of time bar); Johnson v. Rivera, 272 F.3d 519, 522 (7th Cir.2001) (allowing stays in prisoner-rights suits to permit exhaustion without risk of statute-of-limitation bar). Moreover, this procedure comports with Federal Rule of Civil Procedure 23(c)(1)(A), which permits district courts to wait until “an early practicable time” before ruling on a motion to certify a class. We remind district courts that they must engage in a “rigorous analysis” — sometimes probing behind the pleadings — before ruling on certification. Wal-Mart [*897] Stores, Inc. v. Dukes, — U.S. -, 131 S.Ct. 2541, 2551, 180 L.Ed.2d 374 (2011). Although discovery may in some cases be unnecessary to resolve class issues, see 3 Alba Conte & Herbert B. Newberg, New-berg on Class Actions § 7.8, at 25 (4th ed.2002), in other cases a court may abuse its discretion by not allowing for appropriate discovery before deciding whether to certify a class, see Pitts, 653 F.3d at 1093 n. 5; Mills v. Foremost Ins. Co., 511 F.3d 1300, 1311 (11th Cir.2008); Duke v. Univ. of Tex. at El Paso, 729 F.2d 994, 996-97 (5th Cir.1984).

Damasco also contends that the result here would differ if Clearwire had made its offer under Rule 68. He notes that some district courts in this circuit have allowed plaintiffs to avoid mootness by seeking class certification within that rule’s time-frame for accepting or rejecting an offer. See, e.g., Wilder Chiropractic, Inc. v. Pizza Hut of S. Wis., Inc., 754 F.Supp.2d 1009, 1019 (W.D.Wis.2010). But we need not address the propriety of that approach here. Clearwire made its offer while this suit was in state court, and Illinois procedure has no analog to Rule 68.

Damasco next argues that, even if we refuse to create a new exception to mootness, his situation falls under the established exception for inherently transitory claims. But not even the circuits that disagree with us about Damasco’s first argument go so far as to say this type of situation gives rise to an inherently transitory claim. See Pitts, 653 F.3d at 1091; Weiss, 385 F.3d at 347. For a claim to be inherently transitory, uncertainty must exist over whether “any member of the class would maintain a live controversy long enough for a judge to certify a class.” Olson v. Brown, 594 F.3d 577, 580 (7th Cir.2010). As we have discussed, any class member following in Damasco’s footsteps can avoid the barrier he now faces simply by moving to certify a class when filing suit. We discern no other obstacle that would moot a case like Damasco’s before a judge could rule on certification.

Finally, Damasco argues that the district court’s denial of his Rule 59(e) motion is “irreconcilable” with its comments during the Fahey hearing. We review a denial of a motion to reconsider for abuse of discretion. Heyde v. Pittenger, 633 F.3d 512, 521 (7th Cir.2011). Because the court initially remarked that the offer in Fahey did not moot that case, Damasco contends that Clearwire’s offer should not have mooted his case. But as the district court later observed, the focus of the hearing in Fahey was to determine whether a settlement agreement had been reached, not to decide whether the case was moot. The court’s initial, off-the-cuff comments about the existence of mootness are subject to de novo review in this court, see Breneisen v. Motorola, Inc., 656 F.3d 701, 706 (7th Cir.2011), and we have already explained why Clearwire’s offer, which preceded any motion for class certification, mooted this case. The court thus did not abuse its discretion in denying reconsideration.

Affirmed.