Theorem – |s_n – s| <= k*|A_n|

Contents
1. Theorem
2. Proof

Theorem

Let $(s_n)$ and $(a_n)$ be sequences of real numbers and let $s\in{\mathbb{R}}$ . If for some $k>0$ and some and some $m\in \mathbb{N}$ , we have that

$|s_n -s| \leq k|a_n|, \quad \forall n>m$

and that $(a_n) \rightarrow \infty.$ , then $lim_{n\rightarrow \infty}~{s_n} = s$

Proof

The proof follows by letting $|a_n| = \epsilon$