Let $f,g:[0,\infty) \rightarrow \mathbb{R}$ are continuous functions.
Moreover, $f^2$ and $g^2$ are integrable on $[0, \infty)$.
I want to prove uniform convergence of
$$u_m=\int_{0}^{m} f(x+y)g(y)dy \to u=\int_{0}^{\infty} f(x+y)g(y)dy$$
on $[0,\infty)$ as $m \to \infty$.
I already proved that $u$ is bounded on $[0,\infty)$ and $u_m$ are uniformly continuous on every compact subset of $[0,\infty)$ by Cauchy-Schwarz inequality.
However, I do not know what to do from here.
Why doesn't the application give you uniform convergence? It seems like you get the following:
$$ \vert u-u_m\vert \leq \Big\vert \int_m^\infty f(x+y)g(y)dy \Big\vert\leq \sqrt{\int_m^\infty f(x+y)^2 dy } \cdot \sqrt{\int_m^\infty g(y)^2 dy } \\ = \sqrt{\int_{m+x}^\infty f(y)^2 dy } \cdot \sqrt{\int_m^\infty g(y)^2 dy } \leq \sqrt{\int_m^\infty f(y)^2 dy } \cdot \sqrt{\int_m^\infty g(y)^2 dy } . $$
The last terms are independent of $x$ and tend to zero as $m\to \infty$. This gives you uniform convergence.