Real Analysis: product of convergent sequences

[uva123]

Homework Statement

suppose {a_n} and {b_n} are sequences such that {a_n} converges to A where A does not equal zero and {(a_n)(b_n)} converges. prove that {b_n} converges.

Homework Equations

What i have so far:
(Note:let E be epsilon)
i know that if {a_n} converges to A and {b_n}converges to B then {(a_n)(b_n)} converges to AB.

Let {(a_n)(b_n)}converge to a limit, call it L. E > 0 is given, there exists a positive integer N such that n>N implies
|(a_n)(b_n) − L| < E

The Attempt at a Solution

how can i prove that if the product of two sequences is a convergent sequence, then the two multiplies sequences are also convergent? i think i need to prove this with a contradiction but i don't know why if {a_n} is convergent, {b_n} can't be divergent...what does it mean if it is divergent?
please help any way you can!

 

[lanedance]

for proof by contradiction, start with the definition of convergence and take the logical NOT of the statement