Could someone help me prove the following:

If lim (an) = 0 and bn is a bounded sequence, prove lim (anbn) = 0.

Both limits are to infinity.

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- Oct 8th 2006, 01:46 PMJimmyTLimit Proof
Could someone help me prove the following:

If lim (an) = 0 and bn is a bounded sequence, prove lim (anbn) = 0.

Both limits are to infinity. - Oct 8th 2006, 02:04 PMCaptainBlack
As b_n is bounded there exists a b>0, such that |b_n|<b.

Also as lim_{n->infty} a_n=0, for any epsilon>0, there exits an N, such that

|a_n|<epsilon for all n>N.

Hence |a_n b_n|<epsilon b, for all n>N.

So for any epsilon', set epsilon = epsilon' /b, then there exists an N

such that |a_n| < epsilon, and also |a_n b_n|<epsilon' , hence:

lim_{n -> infty} a_n b_n = 0.

RonL - Oct 8th 2006, 02:09 PMJimmyT
Thanks a lot.