Real numbers/Sequences/Squeeze criterion/Fact

Squeeze criterion

Let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} },\,{\left(y_{n}\right)}_{n\in \mathbb {N} }$ and ${}{\left(z_{n}\right)}_{n\in \mathbb {N} }$ denote real sequences. Suppose that

$x_{n}\leq y_{n}\leq z_{n}{\text{ for all }}n\in \mathbb {N}$

and that ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ and ${}{\left(z_{n}\right)}_{n\in \mathbb {N} }$ converge to the same limit ${}a$.

Then also ${}{\left(y_{n}\right)}_{n\in \mathbb {N} }$ converges to ${}a$.

Write a proof