Assume {an} is bounded and divergent and {bn} is a convergent sequence. Show {an.bn} is convergent if and only if {bn} converges to 0.
Added by Heather E.
Step 1
First, we need to show that if {an.bn} is convergent, then {bn} converges to 0. Since {an} is bounded, there exists a constant M > 0 such that |an| ≤ M for all n. Suppose {an.bn} converges to L. Then, for any ε > 0, there exists a positive integer N such that for Show more…
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