(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Assume that (an) is a bounded (but not necessarily convergent) sequence, and that the

sequence (bn) converges to 0. Prove that the sequence (anbn) converges to zero.

2. Relevant equations

3. The attempt at a solution

Assume that an is a bounded sequence and bn converges to 0.

That means for all n in N, there exists a M >0 so that

|an|<=M

Since bn converges, that means that it must be bounded as well. Which means for all n in N there exists a P>0 so that

|bn|<=P

since |an|<=M and |bn|<=P that means for all n in N:

|an||bn|<= MP which is equivalent to |anbn|<=MP

where MP>0 since M>0 and P>0. Hence (anbn) is bounded

Since bn converges to 0 that means for e>0 there exists an N in N so that for n>=N

|bn-0|<e

which is equivalent to -e<bn<e

This is where I get stuck. Do I just multiply the inequality by an? cause then I'd have

-e(an)<bnan<e(an)

which would be equivalent to |anbn|<e2 if I let e2=e(an) which would mean that anbn converges to zero as well. But I don't know if I can multiply the sequence by it though....

Any help would be great!

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# Converging analysis proof

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