We write $lim_{n \rightarrow \infty} s_n = L$ provided that for every positive integer $m$ there is a real number $N$ so that $|s_n-L<1/m$ whenever $n \geq N$.
Definition 2.6: Let ${s_n}$ be a sequence of real numbers. We say that $s_n$ converges to a number $L$ and write $lim_{n \rightarrow \infty}s_n=L$ or $s_n \rightarrow L \ as\ n \rightarrow \infty$ provided that for every number $\epsilon$> 0 there is an integer $N$ so that $|s_n - L| < \epsilon$ whenever $n \geq N$.
To show equivalency, can we just say $1/m = \epsilon$ since we can make $\epsilon$ sufficiently small? It can't be that simple.