real analysis - Prove $limlimits_{nrightarrow infty}x_n=0$ iff $limlimits_{nrightarrow infty}frac{1}{x_n}=infty$

For $x_n \gt 0$ $\forall n=1,2,3,...$ Prove $$\lim\limits_{n\rightarrow +\infty}x_n=0$$ iff $$\lim\limits_{n\rightarrow +\infty}\frac{1}{x_n}=+\infty$$
Working proof:

$\forall M>0, \exists N$ such that $x_n>M$, $\forall n \ge N$

$x_n>M\Rightarrow(1/x_n)<(1/M)$

Choose $N=[M]+1$ $\Rightarrow$ $\frac{1}{x_n}<(1/x_N)=(1/x_{[M]+1})$

I have no idea what the next step would be or if I'm even headed in the right direction...

Answer

Suppose $x_n \to 0$. Let $M$ in $\mathbb R+$, there exists $N$ such that for all $n ≥ N, |x_n| ≤ 1/M,$ so for all $n ≥ N, |1/x_n| ≥ M$ (since $x > 0 \to 1/x$ decreases). So $1/x_n \to +\infty$

The same for the other direction.