While reading about currents I came across the following lemma in Lectures on Geometric Measure Theory by Leon Simon on page 135:
Lemma. If $\left\{T_j\right\}_{j\in\mathbb{N}}$ is a sequence of currents in $\mathcal{D}_k(U)$ such that $$\sup_{j\in\mathbb{N}}\mathbb{M}_W(T_j)<\infty$$ for all $W\subset\kern-2px\subset U$, then there is a subsequence $\{T_{j_i}\}_{i\in\mathbb{N}}$ and a $T\in\mathcal{D}_k(U)$ such that $T_{j_i}\rightharpoonup T$ in $U$ as $i\to\infty$.
For some context, the space $\mathcal{D}_k(U)$ is the space of $k$-dimensional currents on the open set $U\subseteq\mathbb{R}^n$, i.e. the (topological) dual space of the space of smooth compactly supported differential $k$-forms on $U$, $\mathcal{D}^k(U)$. Furthermore, for a current $T\in\mathcal{D}_k(U)$ and a set $W\subset\kern-2px\subset U$, we have that
$$\mathbb{M}_W(T)=\sup\left\{T(\omega) : \omega\in\mathcal{D}^k(U),\,\lVert\omega\rVert\leq1,\,\operatorname{supp}\omega\subseteq W\right\}$$
is the mass of $T$ in $W$. My concern is with how the above theorem is actually proven. In particular, right after defining the notion of weak convergence of currents, i.e. that $T_j\rightharpoonup T$ in $U$ as $j\to\infty$ iff $\lim_{j\to\infty}T_j(\omega)=T(\omega)$ for all $\omega\in\mathcal{D}^k(U)$, it is quickly mentioned that mass is lower semi-continuous with respect to weak convergence (which I can easily verify), and right after that the lemma is given, with the only "proof" being that it's an application of the Banach-Alaoglu theorem on the Banach spaces
$$\mathcal{M}_k(W)=\left\{T\in\mathcal{D}_k(W):\mathbb{M}_W(T)<\infty\right\}$$
for $W\subset\kern-2px\subset U$. Being very inexperienced with functional analysis, I am unable to really make any substantial progress on actually verifying this lemma. While it might be trivial for someone who is more familiar with applications of the Banach-Alaoglu theorem, the only thing I've thought to is to, for all $W\subset\kern-2px\subset U$, let $\alpha_W>0$ be such that
$$\mathbb{M}_W(T_j)\leq\alpha_W$$
for all $j\in\mathbb{N}$, then define a new sequence $\{R^W_j\}_{j\in\mathbb{N}}$ by setting $R^W_j=\frac{T_j}{\alpha_W}$, so that
$$\mathbb{M}_W(R^W_j)\leq1.$$
If I understand Banach-Alaoglu correctly (which I might not), then I should be able to extract a convergent subsequence $\{R^W_{j_i}\}_{i\in\mathbb{N}}$ converging weakly in $W$ to some $R^W$, from which I should be able to deduce that $T_{j_i}\rightharpoonup \alpha_W R^W$ in $W$ as $i\to\infty$. The problem here is that each such subsequence depends on the choice of $W$, and the subsequences only converge weakly on the respective $W$, which is not what the lemma entails. Perhaps what I have concluded (assuming it is correct) will imply the lemma, however I doubt it.
I have also tried looking for other resources containing the same lemma, however each one I was able to find simply stated that it is a consequence of the Banach-Alaoglu theorem without elaborating further.