Inequality Rule for Real Sequences/Proof 2

Proof
Consider the sequence $\sequence {z_n}$ defined by:
 * $z_n := y_n - x_n$

From Sum Rule for Real Sequences:
 * $z_n \to m - l$ as $n \to \infty$

Furthermore, the assumption that $x_n \le y_n$ for all $n \in \N$ means that:
 * $\forall n \in \N: z_n \ge 0$

Applying the Lower and Upper Bounds for Sequences to the sequence $\sequence {z_n}$ leads to the conclusion that $m - l \ge 0$.

That is:
 * $l \le m$