If look

here.
As a result we have a simple result.

**Theorem 1:** Let $\displaystyle (x_n)$ be a convergent sequence of non-negative terms. Then its limits is non-negative. (This should be obvious from the link I gave you).

**Theorem 2:** Let $\displaystyle x_n\geq y_n$ for all $\displaystyle n$. And $\displaystyle (x_n) \mbox{ and }(y_n)$ are convergent sequences. Then $\displaystyle \lim \ x_n \geq \lim \ y_n$.

**Proof:** Define a sequence $\displaystyle z_n = x_n - y_n$. Now this means that $\displaystyle z_n \geq 0$. And furthermore $\displaystyle (z_n)$ is convergent since it is a difference of two convergent sequences. Thus, $\displaystyle \lim \ z_n \geq 0$. Thus, $\displaystyle \lim \ x_n - \lim \ y_n \geq 0$ because the limit of $\displaystyle z_n$ is the difference of the two individial sequences. Thus, $\displaystyle \lim \ x_n \geq \lim \ y_n$ by Theorem 1. Q.E.D.