Convergence proof...

I am a little stuck on the following:
suppose that:
1) sum(n..infinity) (a_n)^2 and sum(n..infinity)(b_n)^2 converges, then prove sum(n..infinity) (a_n)*(b_n) converges.

I am guessing it has something to do with the comparison test but i can't quite put my finger on it, something with 0<=(abs(a_n)-abs(b_n))^2

In addition am i on the right lines with thinking that, as the sum of (b_n)^2 converges, there must exist a number N such that sum(b_n)^2<1 for all n>=N
Then look at the sum that goes from N to infinity of a_n*b_n ? this is the part im stuck at,
any help would much appreciated, many thanks.

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Quote:

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Originally Posted by breitling

I am a little stuck on the following:
suppose that:
1) sum(n..infinity) (a_n)^2 and sum(n..infinity)(b_n)^2 converges, then prove sum(n..infinity) (a_n)*(b_n) converges.

I am guessing it has something to do with the comparison test but i can't quite put my finger on it, something with 0<=(abs(a_n)-abs(b_n))^2

In addition am i on the right lines with thinking that, as the sum of (b_n)^2 converges, there must exist a number N such that sum(b_n)^2<1 for all n>=N
Then look at the sum that goes from N to infinity of a_n*b_n ? this is the part im stuck at,
any help would much appreciated, many thanks.

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Setting it is clear that converges. But because is also converges...

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Marry Christmas from Serbia

Many thanks to both of you