In his influential 1821 textbook Cours d'Analyse, Augustin Cauchy states a theorem that is now widely regarded as false, attributed to a confusion between pointwise convergence and uniform convergence. This mistake — if indeed it was a mistake — is of both pedagogical and philosophical-historical interest.
Counterexamples (specific Fourier series) were known already to Joseph Fourier, and Niels Abel specifically pointed them out as counterexamples to Cauchy's claim. However, Cauchy denied that these were counterexamples, on the grounds that the series did not converge everywhere (which is a hypothesis in the theorem). Imre Lakatos? has argued that the confusion rests on different conceptions of the continuum, so that Cauchy's notion of convergence everywhere is really more like Weierstrass's notion of uniform convergence than pointwise convergence, and the theorem as he intended it is true.
Although Cauchy's original formulation was about the sum of an infinite series, we will consider it in the slightly more elementary context of the limit of an infinite sequence; the modern isomorphism between these was already well established and used explicitly by Cauchy.
Let be an infinite sequence of continuous functions from the real line to itself. Suppose that, for every real number , the sequence converges to some (necessarily unique) real number , defining a function ; in other words, the sequence converges pointwise to . Then is also continuous.
Let be an infinite sequence of continuous functions from the real line to itself. Suppose that the sequence converges uniformly to a function . Then is also continuous.
Let be an equicontinuous sequence of (necessarily continuous) functions from the real line to itself. Suppose that, for every real number , the sequence converges to some (necessarily unique) real number , defining a function ; in other words, the sequence converges pointwise to . Then is also continuous.
The first counterexamples to Non-Theorem arose as Fourier series. The sawtooth wave
may be the simplest. Each partial sum of this infinite series is continuous; the sum converges pointwise as indicated for not a multiple of and to (which is the average of the limits on either side) for a multiple of . However, the sequence of partial sums is not equicontinuous, nor does it converge uniformly. And indeed, the sum is not continuous at multiples of .
A simple counterexample from outside of this context is
on . (We can make this a function on all of by letting be for and letting be for .) Each is continuous; the pointwise limit is for but for , which is not continuous. Again, the sequence is not equicontinuous, and its convergence is not uniform.
Another counterexample is
(The absolute value is here only to handle negative values of ; the example is analytic on .) Now , while for .
Here is Cauchy's argument:
At any , we have
As becomes infinite and becomes infinitesimal, all of the terms on the right become infinitesimal (respectively, because converges to at , because is continuous at , and because converges to at ). Thus, the expression on the left also becomes infinitesimal, proving that is continuous at .
Nowadays, we'd say that a variable approaches zero rather than that it becomes infinitesimal. If we interpret this argument in modern analysis, then the proof is flawed, because (the lower bound for) depends on in the first term, while (the upper bound for the absolute value of) depends on in the second term; they cannot be chosen simultaneously.
Analysing this argument, Philipp von Seidel (first, and others afterwards) realised that it could be fixed if converges to uniformly, so that is independent of . Another fix is to make independent of , by requiring the family to be equicontinuous, although this idea came later.
Writing Cauchy's argument in the epsilontic language developed by Karl Weierstrass, we have:
Let be a positive number, and consider . Because converges to at , there is some natural number such that whenever . Because is continuous at for each , there is some positive number such that whenever . Because converges to at for each , there is some natural number such that whenever . Therefore, there are and such that
whenever and . Fixing any such , is continuous.
We have used the variable names and in two different contexts each, then pretended that they arose in a single context for the final inequality. The error can be made more explicit by using different variable names:
Let be a positive number, and consider . Because converges to at , there is some natural number such that whenever . Because is continuous at for each , there is some positive number such that whenever . Because converges to at for each , there is some natural number such that whenever . Therefore, there are and such that
whenever and . Fixing any such , is continuous.
The final inequality is now clearly spurious, since need not be equal, nor .
This can be fixed up to a point. We may let be and let be , but then we have no control over . Conversely, we may let all be (anything bounded below by) , but then we have no control over . Indeed, Non-theorem is false, as the counterexamples show.
However, the weaker theorems with strengthened hypotheses are valid, by similar proofs:
Let be a positive number, and consider . Because converges uniformly to , there is some natural number such that for every (including ) whenever . Because is continuous at for each such , there is some positive number such that whenever . Therefore, there is such that, whenever , there is such that
whenever . Fixing any such , is continuous.
Let be a positive number, and consider . Because converges to at , there is some natural number such that whenever . Because is equicontinuous at , there is some positive number such that for every whenever . Because converges to at for each such , there is some natural number such that whenever . Therefore, there are and such that, whenever , there is such that
whenever . Fixing any such , is continuous.
Writing Cauchy's argument in the language of nonstandard analysis developed by Abraham Robinson, we have:
Because converges to at , is infinitesimal whenever is infinite. Because is continuous at for each finite , it is continuous at for each infinite (by the transfer principle), so is infinitesimal whenever is infinitesimal. Because converges to at each standard point?, converges to at the nonstandard point (by the transfer principle), so is infinitesimal whenever is infinite. Therefore,
is also infinitesimal whenever is infinite and is infinitesimal. Fixing any such , is continuous.
Both uses of the transfer principle are illegitimate, since the statements to which they're applied are not first-order in the language of the real numbers. Indeed, consider the counterexample near . Then is infinitesimal as claimed iff is infinitesimal, which may fail. Conversely, (for ) is infinitesimal iff is infinite, which may also fail. In between, when is finite and finitesimal, both fail!
We can rewrite this proof more carefully, putting down what really follows in place of the misuse of the transfer principle:
Because converges to at , is infinitesimal whenever is infinite. Because is continuous at for each finite , there is some infinite such that is continuous at for all , so then is infinitesimal whenever is infinitesimal. Because converges to at each standard point? and is nearstandard?, there is an infinite number such that is infinitesimal whenever . Therefore,
is also infinitesimal whenever , , and is infinitesimal. Fixing any such , is continuous.
This is more clearly wrong, since there may be no such in the last step (as may be larger than ).
However, we can modify this to produce correct proofs of the weaker theorems:
Because converges to at , is infinitesimal whenever is infinite. Because is continuous at for each finite , there is some infinite such that is continuous at for all , so then is infinitesimal whenever is infinitesimal. Because converges to uniformly, converges to at the nonstandard point , so is infinitesimal whenever is infinite. Therefore,
is also infinitesimal whenever is infinite and is infinitesimal. Fixing any such , is continuous.
Because converges to at , is infinitesimal whenever is infinite. Because is equicontinuous at , is continuous at even for infinite , so is infinitesimal whenever is infinitesimal. Because converges to at each standard point? and is nearstandard?, there is an infinite number such that is infinitesimal whenever . Therefore,
is also infinitesimal whenever and is infinitesimal. Fixing any such , is continuous.
(To check that these are valid, one must know the nonstandard formulations of uniform convergence and equicontinuity that are used; but they are correct. Specifically, a sequence of functions converges uniformly to a function iff, for every hyperreal number , the sequence of hyperreal numbers converges to the hyperreal number ; and is equicontinuous at a real number iff, for every hyperinteger , the hyperfunction is continuous at .)
On the other hand, in the light of nonstandard analysis, we can also reevaluate Cauchy's original claim. Cauchy himself, when confronted with counterexamples such as the Fourier series, denied that the sequence of functions converged everywhere. One interpretation of this is that it fails to converge at some nonstandard points. By this analysis, Cauchy's notion of convergence everywhere is our modern notion of uniform convergence, not pointwise convergence, and his theorem is true (and his proof correct, even if less detailed than one might like).
In his experimental textbook Proofs and Refutations?, Imre Lakatos? used Cauchy's sum theorem to motivate the concept of uniform convergence. Two years later, he partially reevaluated his discussion in light of the question of what Cauchy's conception of the continuum was.
According to Lakatos, it is ahistorical to interpret Cauchy's 1821 result using either Weierstrass's epsilontics or Robinson's nonstandard analysis. Cauchy did not mean that converges (in ) for each fixed standard real number , nor that it converges for each fixed hyperreal number ; rather, he said that it converges for each variable real number. In particular, when discussing the Fourier series
Cauchy states (in 1853) that the sequence
of partial sums fails to converge when ; that is, is a variable whose value depends on the position in the sequence! Such a claim is hard to interpret in an epsilontic framework; but in a nonstandard framework, it simply means that for some particular infinite values of and infinitesimal values of (namely those for which ), this partial sum is not infinitely close to (which is the value of the partial sum at ), demonstrating that the sequence is not equicontinuous.
If we (still arguably ahistorically) interpret convergence to mean to that converges (in ) for every convergent sequence of real numbers, then this is equivalent (given that each is continuous) to uniform convergence, and Cauchy's result would again be true and his argument would be (when formalised with either epsilontics or nonstandard analysis) valid.
On the other hand (and this goes beyond what Lakatos wrote), one way to interpret Cauchy's claim that (1) fails to converge when is, using nonstandard analysis, simply that for any particular infinite value of and using the infinitesimal value , this partial sum is not infinitely close to (which is the value of the infinite sum at ). This suggests interpreting convergence to mean that is infinitely close to whenever is infinite and is infinitesimal (for real), which again is equivalent (given continuity of ) to uniform convergence, again justifying Cauchy's proof. However, proving that uniform convergence is equivalent to this is basically the same concept as the nonstandard proof of Theorem anyway.
The original sum theorem is in
Cauchy's most thorough defence of the theorem against critics is in
The reconstruction in terms of nonstandard analysis is in the historical appendix to
Lakatos's discussion forms Chapter 3 of
Last revised on May 31, 2022 at 05:58:35. See the history of this page for a list of all contributions to it.