Answer to Question #158014 in Calculus for Vishal

Question #158014

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.


1
Expert's answer
2021-01-29T11:45:47-0500

"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."


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