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(a) Assume $f$ and $g$ are continuous and positive. Prove that if $\int_0^\infty f(x)dx$ converges, and $g(x)$ is bounded, then $\int_0^\infty f(x)g(x)dx$ converges.

(b) Assume $f$ and $g$ are continuous and positive. Prove that if $\int_0^\infty f(x)dx$ and $\int_0^\infty g(x)dx$ converge, this does not imply $\int_0^\infty f(x)g(x)dx$ converges.

I don't see how I may use the boundedness on convergence of the composition product, or how that result would differ from the second question.

Any help is appreciated!

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In (a) we have $f(x) g(x) \leq M f(x)$ for all $x \geq 0$, where $$M:= \sup_{y \in [0,\infty)} g(y) < \infty$$ by assumptation. Note that we have used that both functions are non-negative. This shows already the integrability of the product $fg$.

In (b) we can take $f(x) = g(x)$, where $$g(x) = \sum_{k=1}^\infty 2^k 1_{[k,k+2^{-2k}]}(x).$$ This function is integrable, but $g(x)^2$ is not.

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