Let $y_n=\dfrac{1}{x_n}$

We want to show that $\displaystyle{\lim_{n\to \infty}}x_n=0 \Rightarrow \displaystyle{\lim_{n\to \infty}}y_n=\infty$

Let $M>0$

Since $\displaystyle{\lim_{n\to \infty}}x_n=0$ we know $\exists N \in \mathbb{N} \ni n>N \Rightarrow x_n<\dfrac{1}{M}$

and thus $n>N \Rightarrow y_n=\dfrac{1}{x_n}>M$ and thus $\forall M>0 ~\exists N \in \mathbb{N} \ni y_n>M$

and thus $\displaystyle{\lim_{n\to \infty}}y_n=\infty$