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!
a_{n} bounded look alright As b_{n} is convergent, I would say for any P>0, there exists N such that for all n>N then |b_{n}-0|< |b_{n}| i think you were almost there... now what you need to show to prove a_{n}.b_{n} converges to zero, is that for any e>0 you can choose N, such that for all n>N you have |a_{n}.b_{n}|<e as you know a_{n}<=M for all n, then |a_{n}.b_{n}|<=|M.b_{n}| so now you just need to show you can choose N such that for all n>N |b_{n}|<=e/|M| and i think you're there