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 $n \to \infty$ Let $\left\langle{y_n}\right\rangle$ be another sequence in $\R$.

Let $y_n \to m$ 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 Limits 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, $l \le m$.