calculus - Given: $limlimits_{ntoinfty}a_n=0$ prove $limlimits_{ntoinfty}frac{1}{a_n}=infty$

Please help me to prove $\displaystyle\lim_{n\to\infty}a_n=0$ then $\displaystyle\lim_{n\to\infty}\frac{1}{a_n}=\infty$
Please give me a hint, not a full solution.

I know how to prove $a_n\to\infty \Rightarrow \frac{1}{a_n}\to0$, but not the other way around.

The original problem:
Given $\forall a\in\left\{ a_n \right\}, a<0$ and $\displaystyle\lim_{n\to\infty}a_n=0$ prove: $\displaystyle\lim_{n\to\infty}\frac{1}{a_n}=-\infty$

Answer

Since $\{a_n\}$ is negative and $a_n\to 0$, for each $M\in\mathbb{N}$ there exists $N$ such that $-\frac{1}{M}

Therefore $\frac{1}{a_n}<-M$ for all $n\geq N$, which implies that $\frac{1}{a_n}\to-\infty$.