Is the support of the differential of a k-form a subset of the support of the k-form? I'm finishing up a self-study of Munkres's Analysis on Manifolds and ran across this claim:

Let $\omega$ be a $(k-1)$-form defined on an open set of $R^{n}$ containing a compact oriented $k$-manifold $M$ (for $k>1$). Then the support of $d\omega$ is contained in the support of $\omega$.

Is there a simple way to prove this?
Having a hard time treating the case where $x \in R^{n}$ is a limit point of the support of $d\omega$, rather than simply a point of $R^{n}$ at which $d\omega(x)$ is non-zero.
Thanks in advance.

Here is a proof.
Note that if $\omega$ is a differential form, which is $0$ on some open set $U$, then $d\omega=0$ on $U$ as well (the answer by Albert essentially shows this).
Now let $p$ be a point outside $\text{supp}(\omega)$, and $U$ an open set around $p$ that is disjoint from $\text{supp}(\omega)$.  Then $\omega=0$ on $U$, and by the previous paragraph, $d\omega=0$ on $U$ as well. Thus $p$ is also not in $\text{supp}(d\omega)$.  It's easy to see this implies $\text{supp}(d\omega) \subset \text{supp}(\omega)$.
