Let $\varepsilon > 0$ and choose $n$ such that $n > 1 / \varepsilon$. Then
\begin{equation}
\begin{split}
\left| \frac{1}{n} \right|
= \frac{1}{n}
< \varepsilon
\end{split}
\end{equation}
Since $\varepsilon > 0$ was arbitrary, the claim follows due to
R1089: Characterisation of convergent sequences in metric space. $\square$
