Math Help - A Real Analysis Proof

1. A Real Analysis Proof

I need some help with proving a theorem. I really appreciate any kind of help.

Theorem: Suppose [an] is a convergent sequence and sigma un is a convergent positive series, then sigma ((an)^2)(un) is convergent.

The n's in an and un are subscripts.

2. Originally Posted by straman
I need some help with proving a theorem. I really appreciate any kind of help.

Theorem: Suppose [an] is a convergent sequence and sigma un is a convergent positive series, then sigma ((an)^2)(un) is convergent.

The n's in an and un are subscripts.
Maybe I understand it wrong but it seems true, and not really difficult... Since $(a_n)_n$ is convergent, $(a_n^2)_n$ is convergent as well, hence it is bounded: there is $M$ such that $|a_n^2|\leq M$ for all $n$. From there, we deduce $0\leq a_n^2 u_n\leq M u_n$ and you can conclude.