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Let $\sum|g_n|$ converge uniformly in $X$. Suppose there exists $K$ such that $|f_n(x)|\leq K$, for all $n\in\mathbb{N}$ and all $x\in X$. Prove that $\sum f_ng_n$ converges absolutely and uniformly in $X$.

How can I do this problem, it's from a book that I'm using of real analysis, can someone help me with the solution D:? It use a trick or something?

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    You have two uses of $k$ in the question. In $\sum f_kg_k$ if you use $|f_n(x)|\leq k$ to replace $f_k$ by $k$2011-11-17
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    @t.b.: I have satisfied my commitment with an upvote. I was happy to do so. Not that 10 points matters to you. Good answer.2011-11-17
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    @Ross: Thanks!${}{}$2011-11-17
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    I was wondering if you could tell me what's missing in my answer. Also, I noticed that you haven't accepted any answers recently. Please tell the answerers what you expect more or do accept their answers if they were helpful.2012-01-05

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