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 $x_n\ge y_n$ for all $n\in \mathbb{N}$.

Then $\underset{n\to \mathrm{\infty }}{lim}{x}_n\ge \underset{n\to \mathrm{\infty }}{lim}{y}_n$.

Write a proof