This is a particular fact that I ended up proving in the process of attempting one of my recent homeworks, but I don't think I've seen this particular fact online even though it feels like a fairly simple fact and I so I'm not entirely sure if I have made an error somewhere or if I have simply stated some obvious well known fact.
Let $\{u_k\}_{k \in \mathbb N} \in L^2(\mathbb R^n)^\mathbb N$ be such that for any fixed compact set $C$, we have that:$$\int_C|u_k|^2 dx = \|u_k\|_{L^2(C)}^2 < K(C)$$In other words, it is uniformly bounded irrespective of $k$ and only dependent on $C$. Then there exists a subsequence of $u_k$ such that for any compactly supported $\varphi \in L^2(\mathbb R^n)$:$$\int_{\mathbb R^n} \varphi u_k \, dx \to \int_{\mathbb R^n} \varphi u \,dx$$For some $u$ measurable, in other words, $u_k \to u$ weakly in some sense.
The proof idea is pretty simple, knowing that bounded $L^2$ sequences admit a weakly convergent subsequence (Banach-Alaoglu), we may extract a convergent subsequence $\{u_{k_1(n)}\}_{n \in \mathbb N}$ in the weak $L^2$ topology on the ball $B_1$ of radius $1$ at the origin. Repeating this process, we extract a sub-subsequence $\{u_{k_2(n)}\}_{n \in \mathbb N}$ convergent on $B_2$. Iteratively, we extract a subsequence of $\{u_{k_m(n)}\}_{n \in \mathbb N}$, call it $\{u_{k_{m+1}(n)}\}_{n \in \mathbb N}$, so that it is convergent in $L^2$ weakly on $B_{m+1}$. Taking the diagonal sequence now, $\{u_{k_n(n)}\}_{n \in \mathbb N}$ is weakly convergent on every ball of finite radius and thus weakly convergent (the integral against a compactly supported $\varphi$ would converge since $u_k$ is strongly convergent on $\varphi$'s support).
In particular, I'd be very grateful if someone could point out if my proof has an error in it, specifically I'm a bit concerned about whether the limits remain consistent each time a extract a subsequence. Or if someone could point me to a reference of this fact.