Let $a_n$ and $b_n$ be sequences . if ... $\lim _{n\to \infty } |a_n - b_n| = 1$ prove that $b_n$ is Convergent Let $a_n$ and $b_n$ two  sequences .
if $a_n$ is Convergent
and  $\lim _{n\to \infty } |a_n - b_n| = 1$ 

prove that $b_n$ is also Convergent

I know that it is not true but I need to find example I think 
$a_n=\frac{1-n}{n}$
$b_n = \left(-\frac{1}{2}\right)^n$
Is this correct and if not how to prove it ? 
thanks 

Hint: Try $a_n=0$ and $b_n=(-1)^n$.
