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).