If a_n --> 0 and there is a sequence b_n that is bounded, then a_n * b_n --> 0
I am lost on how to prove this.
Since we only know that b_n is bounded, it doesn't necessarily converge or have a limit?
So how do I go about proving that a_n * b_n has a limit?
Thanks!
when proving something, if you have no idea where to start, just go by the definition. what does it mean for the limit to approach something? what is the definition of a limit? what is it that you have to show?
you should know that we say the limit of a sequence exists and equals iff
For every , there is some such that implies
what can you do with that?
when you get a bit more experienced with that approach, then you can use more sophisticated approaches, like the squeeze theorem, which is what i would use here. what can you say about ?