- #1
dancergirlie
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Homework Statement
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.
Homework Equations
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!