Inequality Rule for Real Sequences/Proof 2

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:

$\ds \lim_{n \mathop \to \infty} x_n$

$=$

$\ds l$

$\ds \lim_{n \mathop \to \infty} y_n$

$=$

$\ds m$

Let there exist $N \in \N$ such that:

$\forall n \ge N: x_n \le y_n$

Then:

$l \le m$

Consider the sequence $\sequence {z_n}$ defined by:

$z_n := y_n - x_n$

$z_n \to m - l$ 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:

$l \le m$

$\blacksquare$

The Inequality Rule for Real Sequences is also presented on $\mathsf{Pr} \infty \mathsf{fWiki}$ as: