I am currently reading an article, and there is a passage where I can't understand how to justify what the author says. I'll try to write what I think the relevant informations are, since a precise explanation of the background would be definitely too long.
Let's say we have a sequence of functions $\{u_n\} \subset H^1 (\mathbb{R}^3)$ that are bounded in norm. Thanks to the compact embedding theorem, I can say that (up to a subsequence which I'll not rename), there is a function $\bar{u}$ such that
$$ u_n \to \bar{u} \text{ in } L^2 (B(0,1))$$
with $B(0,1)$ the unitary ball in $\mathbb{R}^3$. Now he basically says that this is enough to have a dominating function in $L^2 (B(0,1))$, indeed he later applies Lebesgue dominated convergence. But why is this true? How? I can't see how to construct or prove the existence of this dominating function.