Limit of sequences with the squeeze theorem with sequences

Question

If $\lim x_n$ and $\lim y_n$ exists and if $x_n\leq w_n\leq y_n$ for each $n\in\mathbb{N}$ then $\lim w_n$ also exists.

Initially, I assumed that this is a weaker statement of the squeeze theorem for sequences and that it was true, but have learned that the statement is false. What about this statement makes it false?

Answer 1

It is false because, for instance, if you take $x_n=-1$, $y_n=1$, and $w_n=(-1)^n$, you indeed have$$(\forall n\in\mathbb{N}):x_n\leqslant w_n\leqslant y_n,$$but the limit $\lim_{n\to\infty}w_n$ does not exist.