If (b_n) is a bounded sequence and lim(n->infinity) a_n = 0, show that lim(n->infinity) [a_n][b_n] = 0.?

Question

If (b_n) is a bounded sequence and lim(n->infinity) a_n = 0, show that lim(n->infinity) [a_n][b_n] = 0.?

Answer 1

Since it's bounded there is some B>0 such that |b_n| ≤ B for all n.

Let є > 0.  Since lim(n->inf) a_n = 0, there's an N such that n>N makes

|a_n| < є/B.

Then n>N makes |a_n * b_n| = |a_n| |b_n| < є/B * B = є, which is the statement that lim(n->inf)[a_n*b_n]=0.

Answer 2

Bn bounded means there exists some positive M such that

|Bn| < M for all n.

To prove that lim AnBn is zero, we need to show that given any ε > 0, there is some N such that n > N implies that

| AnBn - 0 | < ε , or |AnBn| < ε

So assume we are given some ε.

Since lim An = 0, it means that there exists some N such that n > N implies that

  | An - 0 | < ε/M , or  |An| < ε/M.

Now consider | AnBn - 0| = |AnBn| = |An||Bn|

which is <= (ε/M)* M = ε  when n > N. In other words, AnBn converges to zero.

  

Answer 3

using a common theorem that applies when both sequences converge;

lim {an}{bn} = lim {an} * lim {bn}

n -> infinity

if lim {an} -> 0

then lim {an}{bn} = 0*lim {bn} = 0