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The following is exercise 2.3.3 (Squeeze Theorem) from Stephen Abbott's Understanding Analysis 2nd.

Show that if $x_n ≤ y_n ≤ z_n$ for all $n∈N$, and if $\lim x_n = \lim z_n =l$, then $\lim y_n =l$ as well.

My solution appears to be much simpler than an answer I have found in a solutions manual which uses the triangle inequality. Is my solution flawed?

First we write out the given conditions in terms of the definition of a limit. In the following $\epsilon >0$, and $n \geq N$ for some $N$, as usual.

$$ -\epsilon < x_n - l < \epsilon $$ $$ -\epsilon < z_n - l < \epsilon $$

Since the following are also given

$$ y_n \leq z_n $$ $$ x_n \leq y_n $$

We have

$$ y_n - l < \epsilon $$ $$ -\epsilon < y_n - l $$

Which we can combine to the desired

$$ -\epsilon < y_n - l < \epsilon $$

for $n$ sufficiently large. That is, $\lim y_n = l$.

Other questions don't answer this specific question (eg this one, this one, and this one).

Penelope
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    It looks fine to me. – José Carlos Santos Aug 06 '23 at 14:57
  • Yup, that's completely fine. An alternative simple proof is, for any $n\in\Bbb N$ we have: $$|y_n-\ell|\le y_n-x_n+|x_n-\ell|\le z_n-x_n+|x_n-\ell|\le|z_n-\ell|+2|x_n-\ell|$$ and the right hand side vanishes as $n\to\infty$. There are many similar ways to do this – FShrike Aug 06 '23 at 15:08
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    This lesson from this is that a book, even one written by a distinguished author, does not always have the best (shortest, simplest, etc.) proof. I've seen proofs that take pages in a book but can be replaced by single paragraph ones. When you're learning something from a book or paper, I encourage you to try to find your own proofs, before or after you read the published proof. Sometimes, you do find your own, but even if you don't, the effort prepares you well to understand the published one. – Deane Aug 06 '23 at 15:35
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    The shortest proof is given in the first sentence of this post :) – Ryszard Szwarc Aug 06 '23 at 20:44
  • The proof you wrote in question is the one which I am used to. It becomes a bit simpler if one writes inequalities like $l-\epsilon<x_n\leq y_n\leq z_n<l+\epsilon$. – Paramanand Singh Aug 07 '23 at 05:26

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