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I am teaching foundations of analysis, and I would like to have some simple and/or interesting statement to prove in class (or give as homework) where the best way to show that a limit exists is to show that $\liminf = \limsup$.

I know that the standard example in an introductory class is to show that a Cauchy sequence has a limit, but I introduced the real numbers as equivalence classes of Cauchy sequences of rational numbers, so it would end up being a circular and unneeded argument. All the other examples I remember that involve this trick are from more advanced classes (like using Fatou's lemma), so they are not suitable for this class.

Any suggestion?

Stefano
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2 Answers2

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Here is one example that I have in mind since I did it yesterday: the proof that a Darboux-integrable function is Riemann-integrable.

Martin Argerami
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Suppose there are three sequences $a_n,b_n,c_n$ satisfying:

(1) for an arbitrary $\varepsilon>0$, there exists $N$ such that for all $n>N$, $c_n-\varepsilon<a_n<b_n+\varepsilon$;

(2) $b_n,c_n$ both converge to $A$.

Prove that $a_n\to A$ as well.

Proof. Take liminf in $c_n-\varepsilon<a_n$ and take limsup in $a_n<b_n+\varepsilon$ we have $$A-\varepsilon\leq\liminf a_n\leq\limsup a_n\leq A+\varepsilon$$ Since $\varepsilon$ is arbitrary, we have $$\liminf a_n=\limsup a_n=A\implies\lim a_n=A$$ Q.E.D.

Remark. 1. This looks like the squeeze theorem very much. And I think there is an alternative argument similar to the proof of the squeeze theorem to avoid using limsup and liminf, but it will look a lot more uglier.

Remark. 2. If I remember correctly, this is the same as the part in the proof that a Cauchy sequence has a limit. You should note that this technique comes up in other places as well. For example, sometimes you want to determine the limit of a sequence $a_n$ by estimating it when all you have is something like (1), and the convergence of $a_n$ is not a priori known. I remember doing so a few times when solving various problems, but sadly I can't come up with a more specific example off the top of my head.

trisct
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