Lower and Upper Bounds for Sequences/Corollary

Theorem

Let $\sequence {x_n}, \sequence {y_n}$ 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:

$\displaystyle \lim_{n \mathop \to \infty} x_n \le \lim_{n \mathop \to \infty} y_n$

This is often phrased as: limits preserve inequalities.

Proof

Consider the sequence $\sequence {z_n}=$ 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 $\sequence {z_n}$ leads to the conclusion that $m - l \ge 0$.

That is, $l \le m$.

$\blacksquare$