Question

Let {an}∞n=1 be a bounded sequence and {bn}∞n=1 be a sequence converges to 0. Prove that the sequence {an · bn}∞n=1 converges to 0.

Answer 1

$Solution: ~Since~ \{a_n\} ~ is~ a ~bounded~ sequence, there~ exists~ a ~M>0~ such~ that \\| a_n | \leq M~ for ~all ~n \in N. Since~ \{b_n\} ~converges~ to~ zero,~ given ~\epsilon >0, there~exists~ n_0 ~ such~ that~ \\|b_n| < \frac{\epsilon}{M}~for ~ all~ n \geq n_0 \\Now~ |a_n b_n|=|a_n||b_n| < \frac{\epsilon}{M}M= \epsilon~ for ~ all~ n \geq n_0 \\Hence~ \{a_n.b_n\} ~converges ~to ~0.$