Prove divergence of a product of sequences using contradiction If a sequence converges (not to zero) and another sequence diverges, prove by contradiction that the product of these sequences is divergent. 
This is a past test paper question that I am looking at and I really have no idea how to write the proof. My own working doesn't make any sense (even to me...) Would really appreciate some help!! 

If $(a_n)$ converges to $L\neq 0$, then you have $\vert a_n \vert \ge \vert L \vert /2 >0$ for $n$ large enough. Let say $n\ge N$.
Now if the sequence $(b_n)$ diverges, either it is not bounded and therefore $(a_n b_n)$ is not bounded either and therefore diverges. And if $(b_n)$ is bounded (and doesn’t converge), it has at least two limit points $B_1,B_2$. $B_1 L, B_2 L$ are then two limit points of $(a_n b_n)$ that can’t converge.
