Inequality Rule for Real Sequences

Theorem
Let $\sequence {x_n}$ and $\sequence {y_n}$ be sequences in $\R$.

Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent to the following limits:


 * $\ds \lim_{n \mathop \to \infty} x_n = l$
 * $\ds \lim_{n \mathop \to \infty} y_n = m$

Let there exist $N \in \N$ such that:
 * $\forall n \ge N: x_n \le y_n$

Then:
 * $l \le m$