I'm pretty sure this limit is zero, but I was hoping someone could help me show it with algebra:

Suppose that lim as n goes to infinity of (an) = 0. Let (bn) be a non negative sequence with lim as n goes to infinity of (b1 + b2 + · · · + bn−1 + bn) = infinity.

What is the limit as n goes to infinity of: (a1b1 + a2b2 + · · · + an−1bn−1 + anbn)

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(b1 + b2 + · · · + bn−1 + bn)

Should you divide everything by anbn or just bn? I'm not sure.

Thank you for any help.