Let $\Lambda_i$ be a converging sequence of distribution in $D (\Omega)$ for some $\Omega$ open set in $\mathbb{R}^n$. I want to prove that the order of all $\Lambda_i$ is bounded.
Since the sequence converges in $D (\Omega)$, it does the same in $D_{K} (\Omega)$ for each $K$ compact set in $\mathbb{R}^d$.
$D_{K}$ is a Frechet space; moreover, because of the convergence, for each $f \in D_{K}$ the sets in the form $\lbrace \Lambda_i f | i = 1, ..., \infty \rbrace$ are bounded.
Hence by the Banach-Steinhouse Theorem the family $\lbrace \Lambda_i \rbrace$ is equicontinuous when restricted to $D_K$.
The same reasoning applies to $\lbrace D^{\alpha}\Lambda_i \rbrace$, since the convergence of $\Lambda_i$ implies the convergence of all its derivatives.
Now the trouble begins... since all $\Lambda_i$ are continuous, for each compact set $K$ it holds that $|\Lambda_i f| \leq C_K ||f||_{N (K)} $.
I want to find an $N$ such that the previous one holds uniformly in respect to $K$.
I do not have any idea on how to proceed.