Example (again):

$(\frac{n}{n+1})_{n \in \mathbb{N}}$

Choose $\varepsilon > 0$. Find $N$ such that $\vert\frac{n}{n+1} - 1\vert < \varepsilon$

We want: $1 - \frac{n}{n+1} < \varepsilon$
$\frac{1}{n+1} < \varepsilon\\ n+1 > \frac{1}{\varepsilon}\\ n > \frac{1}{\varepsilon} - 1$

$\varepsilon = \frac{1}{10}$ if $n > \frac{1}{\frac{1}{10}} = 10 - 1 = 9$, then $\vert 1 - \frac{n}{n+1}\vert < \frac{1}{10}$