We study approximately stable committee selection under approval preferences, following Cheng et al.~[EC~2019] and Jiang, Munagala, and Wang~[STOC~2020]. In an approval-based committee election, $n$ voters each approve a subset of the candidates, and a committee of size $k$ is selected. A size-$k$ committee is $λ$-stable if, for every nonempty subset $T$ of candidates, the number of voters who strictly prefer $T$ is less than $λ$ times the proportional share $\frac{n|T|}{k}$. We consider sequential Phragmén, a classical polynomial-time voting rule, and prove that it always returns a $2$-stable committee. This implies that the $2$-core is always nonempty and improves upon the $3.651$-stability guarantee of Gao, Sun, and Vondrák~[EC~2026]. The guarantee also extends to weighted voters and is tight even for unweighted elections: for every $1\leλ<2$, sequential Phragmén can return a committee that is not $λ$-stable. Compared with the exact-core algorithm developed in concurrent work by Becker, Greger, and Peters~[arXiv~2026], sequential Phragmén is simpler to implement, and our proof of its $2$-stability is elementary.
We prove that the set $\mathcal{N}_N$ of ReLU networks with one hidden layer and at most $N\in\mathbb N$ neurons is weakly closed in $W^{1,p}(Ω)$, for a bounded connected open set $Ω\subset\mathbb R^d$. Consequently, every coercive, weakly lower semicontinuous energy on this space attains its minimum over $\mathcal{N}_N$. This covers the Deep Ritz method with convex integrands, Sobolev training and Lipschitz learning. Moreover, as the weight decay parameter tends to zero, minimizers of the regularized problem converge, up to subsequences, to minimizers with minimal parameter norm. We also prove that the set of ReLU networks with two hidden layers of bounded widths is not closed in $W^{1,p}(Ω)$ or in $C(\overlineΩ)$. This disproves a conjecture of Petersen, Raslan and Voigtlaender.