Problem Statement: Is the locally uniform limit of locally exact forms locally exact? (That is, if $\omega_{k}\rightarrow \omega$ uniformly on compact subsets of $U$ ($\omega_{k}$, $\omega$ $C^{1}$ $1$-forms in $U$), and each $\omega_{k}$ is locally exact, is the same true for $\omega$?)
I am trying to solve this problem, but I do not know if my approach is leading me to where I need to be or not. I am assuming that the conjecture is true, but this is partly because I haven't been able to come up with a counterexample:
Sketch: Let $U\subset \mathbb{R}^{n}$ be open and connected. Let $(\omega_{k})_{k\in\mathbb{N}}$ be a sequence of locally exact $C^{1}$ $1$-forms in $U$ such that $\omega_{k}\rightarrow \omega$ uniformly on compact subsets of $U$ ($\omega$ a $C^{1}$ $1$-form in $U$). That is, for each $K\subset U$ compact, we have $\omega_{k}\vert_{K}\rightarrow \omega\vert_{K}$ uniformly.
Since each $\omega_{k}$ is locally exact, then for any $x_{0}\in U$, $\exists r>0$ so that $B:=B_{r}(x_{0})\subset U$ with $f_{k}\in C^{1}(B)$ a potential function for $\omega_{k}$ on $B$. That is, $df_{k}\vert_{B}=\omega_{k}\vert_{B}$.
Since each $\omega_{k}\rightarrow \omega$ uniformly on compact subsets of $U$, then for each $K\subset U$ compact, we have for all $\varepsilon >0$, $\exists N=N(\varepsilon)>0$ so that $$\lVert \omega_{k}(x)-\omega(x)\rVert <\varepsilon,\ \ \ \ \forall k\geq N,\ x\in K.$$ Then for each $x_{0}\in \mathrm{int}(K)$, there is some $r>0$ so that $B_{x_{0}}=B_{r}(x_{0})\subset U$ with $f_{k}\in C^{1}(B_{x_{0}})$ a potential for $\omega_{k}$. Then choosing $R=\min\left\{d(x_{0},\partial K), r\right\}$ and $0<\delta<R$, and setting $B_{\delta}=B_{R-\delta}(x_{0})$, it follows that $df_{k}\vert_{B_{\delta}}$ is a potential for $\omega_{k}$ on $B_{\delta}$ $$\lVert df_{k}(x)-\omega(x)\rVert <\varepsilon,\ \ \ \ \forall k\geq N,\ x\in B_{\delta}\subset K.$$
Here is where I am stuck: I must show that for any $y\in U$ there is an $r_{y}>0$ so that $B_{y}:=B_{r_{y}}(y)\subset U$ with $f\in C^{1}(B_{y})$ a potential for $\omega$ on $B_{y}$. That is, $df\vert_{B_{y}}=\omega\vert_{B_{y}}$.
For $k\geq N$ fixed, I was thinking to choose $f=f_{k}$, but that would not make much sense because then $df\neq \omega$, but rather $df\rightarrow \omega$.
I was also told to possibly use the Poincaré Lemma, that if $U$ is starshaped, then any closed $1$-form is exact in $U$ (and a corollary is that a $1$-form is closed if and only if it is locally exact). So maybe I should be trying to show that the limit $\omega$ is closed.
I appreciate any hints or suggestions!