Analysis Sequences Proof and Tending to ∞

The following is false. Disprove it using a counter example.

Claim: If (a_n) → ∞ and b_n> 0 ∀n ∈ N then (a_n b_n) → ∞.

Counterexample:

My attempt.

My pitiful attempt consisted of assuming b_n was a sequence greater than 0. I made a_n be natural numbers (the sequence 1, 2, 3, 4,...) and b_n be the sequence (5, 1/n, 5, 1/n, 5, 1/n,...). Multiplied together it left (5, 1, 15, 1, 25, 1,...) which i thought answered the question as it doesn't tend to infinity.

I thought i had it, but i don't. As b_n isn't in brackets it implies it is a constant from the sequence (b_n), which i can choose. I don't know the sequence (b_n) so it can basically be any number i presume? All i know is it's greater than 0. I can't think of a number for b_n that satisfies this constraint.

Any and all help appreciated, I'm perplexed and out of ideas.