Let $M$ and $N$ be two oriented smooth Riemannian manifolds and consider the product manifold $M\times N$. Suppose that $(\chi_n)_{n\in \mathbb{N}}$ is a sequence of distributions acting on compactly supported test functions $\phi \in C^\infty_0(M\times N)$. If we want to prove that $\chi_n\to \chi$ in the distributional sense, is it enough to consider the distributional action on product functions? In other words, is
$$\lim_{n\to \infty}\int_{M\times N} d^n x \sqrt{g_M(x)} d^m y \sqrt{g_N}(y) \chi_n(x,y) \phi(x)\psi(y)=\lim_{n\to \infty}\int_{M\times N} d^n x \sqrt{g_M(x)} d^m y \sqrt{g_N(y)} \chi(x,y) \phi(x)\psi(y)$$
for $\phi\in C^\infty_0(M)$ and $\psi\in C^\infty_0(N)$ sufficient to conclude that $\chi_n\to \chi$? I feel that the answer is yes, but I'm not sure how to prove.
The reason I think this is the case is because tensor product spaces are spanned by decomposable tensors, so it seems that functions of the form $\phi(x)\psi(y)$ should span $C^\infty_0(M\times N)$. Still, I'm unsure of this statement here because the involved spaces are infinite-dimensional and I worry about convergence issues.