Real Analysis: product of convergent sequences

[slb123]

Problem:
suppose {a[sub:160ax5c9]n[/sub:160ax5c9]} and {b[sub:160ax5c9]n[/sub:160ax5c9]} are sequences such that {a[sub:160ax5c9]n[/sub:160ax5c9]} converges to A where A does not equal zero and {(a[sub:160ax5c9]n[/sub:160ax5c9])(b[sub:160ax5c9]n[/sub:160ax5c9])} converges. prove that {b[sub:160ax5c9]n[/sub:160ax5c9]} converges.

What i have so far:
(Note:let E be epsilon)
i know that if {a[sub:160ax5c9]n[/sub:160ax5c9]} converges to A and {b[sub:160ax5c9]n[/sub:160ax5c9]} converges to B then {(a[sub:160ax5c9]n[/sub:160ax5c9])(b[sub:160ax5c9]n[/sub:160ax5c9])} converges to AB.

Let {(a[sub:160ax5c9]n[/sub:160ax5c9])(b[sub:160ax5c9]n[/sub:160ax5c9])} converge to a limit, call it L. E > 0 is given, there exists a positive integer N such that n>N implies
|(a[sub:160ax5c9]n[/sub:160ax5c9])(b[sub:160ax5c9]n[/sub:160ax5c9]) ? L| < E

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 dont know why if {a[sub:160ax5c9]n[/sub:160ax5c9]} is convergent, {b[sub:160ax5c9]n[/sub:160ax5c9]} cant be divergent....what does it mean if it is divergent?
please help any way you can!!

 

[tkhunny]

Perhaps you could build your "L".

Since the both converge, it is reasonable to assume that each can be very close at the same time. Certainly, we can get closer than unity (1). Then, for some very small deviation from A and B (call them la and lb), we have:

\(\displaystyle (a_{n}+l_{a})*(b_{n}+l_{b})\;=\;a_{n}\cdot b_{n}+l_{a}\cdot b_{n}+l_{b}\cdot a_{n}+Really\;Small\;Quadratic\;Thing\)

What say you? Are we getting anywhere?