# Real numbers/Sequences/Squeeze criterion/Fact/Proof/Exercise

Let and be three real sequences. Let for all and and be convergent to the same limit . Prove that also converges to the same limit .

Let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} },\,{\left(y_{n}\right)}_{n\in \mathbb {N} }$ and ${}{\left(z_{n}\right)}_{n\in \mathbb {N} }$ be three real sequences. Let ${}x_{n}\leq y_{n}\leq z_{n}$ for all ${}n\in \mathbb {N}$ and ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ and ${}{\left(z_{n}\right)}_{n\in \mathbb {N} }$ be convergent to the same limit ${}a$. Prove that also ${}{\left(y_{n}\right)}_{n\in \mathbb {N} }$ converges to the same limit ${}a$.