Lower and Upper Bounds for Sequences/Corollary
Theorem
Let $\left\langle{x_n}\right\rangle, \left\langle{y_n}\right\rangle$ be sequences in $\R$.
Let $x_n \to l, y_n \to m$ as $n \to \infty$
Suppose that for all $n \in \N$, $x_n \le y_n$.
Then $l \le m$, that is, $\displaystyle \lim_{n\to\infty} x_n \le \lim_{n\to\infty} y_n$.
This is often phrased as: limits preserve inequalities.
Proof
Consider the sequence $\left\langle{z_n}\right\rangle$ defined by $z_n := y_n - x_n$.
The Sum Rule for Real Sequences grants that $z_n \to m-l$ as $n \to \infty$.
Furthermore, the assumption that $x_n \le y_n$ for all $n \in \N$ means that $z_n \ge 0$ for all $n \in \N$.
Applying the main result to the sequence $\left\langle{z_n}\right\rangle$ leads to the conclusion that $m-l \ge 0$.
That is, $l \le m$.
$\blacksquare$