Prove that $b_k \to 1/a$ if $a_k \to a$ where $b_0 = 0$ and $b_k = 1/a_k$ for $k>0$ 
Let $a_n$ where $n \in \mathbb {N}$ be a sequence of rational numbers converging to $a$. Suppose $a \neq 0$, for $k = 1, 2, ...$ let
   $$b_k=\begin{cases} 0 & \text{if}\;a_k=0\\\\\frac{1}{a_k} &\text{if}\;a_k \neq 0\end{cases}$$
  Prove that $b_n$ converges to $\frac{1}{a}$.

I was studying real analysis and got stuck on this problem.
Can you help me solve this problem or give me some hints?
Thanks
edit: is it possible to solve this in terms of Cauchy Sequence?

Hint
Because $a\not=0$ (let's say $a>0$), show that $\exists n_0\in\mathbb{N}: \forall n\geq n_0\quad a_n>0$ 
