Real numbers/Convergent sequences/Compare/Fact

Suppose that ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ and ${}{\left(y_{n}\right)}_{n\in \mathbb {N} }$ are convergent sequences such that ${}x_{n}\geq y_{n}$ for all ${}n\in \mathbb {N}$.

Then ${}\lim _{n\rightarrow \infty }x_{n}\geq \lim _{n\rightarrow \infty }y_{n}$.

Write a proof