For an arbitrary vector space $V$ over $\mathbb{F}$, consider continuous linear maps $f: V \to \mathbb{F}$ where continuity is defined as sequential continuity, i.e. if $\phi_j \to \phi$ in $V$ then $f(\phi_j) \to f(\phi)$ in $\mathbb{F}$.
A distribution $u \in \mathcal{D}'(\mathbb{R}^n)$ is a continuous linear functional on $C_C^\infty(\mathbb{R}^n)$ with the above definition of continuity.
In the space $C_C^\infty(\mathbb{R}^n)$, we say that a sequence $\lim_{j \to \infty} \phi_j = \phi$ if there is a compact set $K$ such that $\phi_j \equiv 0$ outside $K$ for all $j$ and all derivatives of $\phi_j$ converge uniformly to the corresponding derivatives of $\phi$.
Now, in the text I'm reading it claims that, with the above definitions, the continuity of $u$ is equivalent to: for all $K$ compact there exists $m$ and $C > 0$ such that for all $\phi \in C_C^\infty(\mathbb{R}^\infty)$ with $\textrm{supp}\, \phi \subset K$, $$ |u(\phi)| \leq C\sum_{|\alpha| \leq m} \sup_{\mathbb{R}^n}|\partial^\alpha \phi|. $$
I'm having some trouble figuring out how these definitions are equivalent. In particular, the definitions of continuity are given in terms of a sequence $\phi_j$ and limiting value $\phi$, but in this equivalent formulation we don't have any sequences. Are they folded into the constant $C$?