Define a sequence of distribution $u_n$.
Let $u_n\to u$ in $D'(X)$ and assume we have a seuqence $\varphi_n\in C^\infty_c(X)$ such that $\varphi_n\to \varphi $ in $C_c^\infty(X)$.
Can we show $$(u_n,\varphi_n) \to (u,\varphi)$$
I know we can show $(u_n,\phi) \to (u,\phi)$ for any $\phi\in C_c^\infty(X)$,and $(u_n,\varphi_j) \to (u_n,\varphi)$ for each $n$.How to combine them togethor?
$$\lim_k\lim_n (u_n,\varphi_k) = (u,\varphi)$$
But not exactly two same variable?
I assume that $X$ is an open subset of $\mathbb{R}^n$. For any compact subset $K$ of $X$, let $C_K^{\infty}(X)$ denote the Frechet space of all $f \in C_c^{\infty}(X)$ such that $\text{supp}(f) \subset K$.
A non-trivial Theorem about convergence in the strict inductive limit topology of $C_c^{\infty}(X)$ implies that there must be $n_0 \geq1$ and a compact subset $K \subset X$ so that each $\varphi_n$ with $n \geq n_0$ and $\varphi$ itself belong to $C_{K}^{\infty}(X)$ and that $\varphi_n \rightarrow \varphi$ in this space. The restriction map $C_{c}^{\infty}(X)^{\ast} \rightarrow C_K^{\infty}(X)^{\ast}$ is continuous for the weak-star topologies and hence the sequence of restricted distributions $u_n|_{C_K^{\infty}}$ converges to the restricted distribution $u|_{C_K^{\infty}}$ in the weak-star topology on $C_K^{\infty}(X)^{\ast}$.
We have thus reduced our problem to proving that in every Frechet space $V$, for every convergent sequence of vectors $\varphi_n \rightarrow \varphi$ and weak-star convergent sequence of continuous linear functionals $\ell_n \rightarrow \ell$, we have $\ell_n(\varphi_n) \rightarrow \ell(\varphi)$ in $\mathbb{C}$, as $n \rightarrow \infty$.
By a further easy reduction, it suffices to prove this in the case $\varphi=0$ and $\ell = 0$.
This in turn follows from the uniform boundedness-principle in Frechet spaces, as explained in this answer. This Theorem implies that the family $\{\ell_n \}_{n \in \mathbb{N}}$ is automatically equi-continuous, meaning that, given any $\varepsilon >0$, there is $U \subset X$ open, $0\in U$, so that for all $(n,v) \in \mathbb{N} \times U$ we have $|\ell_n(v)| < \varepsilon$. So given $\varepsilon$, first choose such such $U$ and then take $n$ sufficiently large so that $\varphi_n \in U$.