It is known that the Schwartz class $\mathcal{S}(\mathbb{R}^n)$ is a Fréchet space and also that the space of test functions $\mathcal{D}(\mathbb{R}^n)$ is dense in $\mathcal{S}(\mathbb{R}^n)$. In particular $\mathcal{D}_K(\mathbb{R}^n) \hookrightarrow \mathcal{S}(\mathbb{R}^n) \hookrightarrow L^1(\mathbb{R}^n)$ are continuous inclusions. Let $\mathcal{S}'(\mathbb{R}^n)$ the topological dual space $\mathcal{S}(\mathbb{R}^n)$. Now we have that the mapping $u \in \mathcal{S}'(\mathbb{R}^n) \longmapsto v=u_{|\mathcal{D}(\mathbb{R}^n)} \in \mathcal{D}'(\mathbb{R}^n)$ is linear and one-to-one because convergence in $\mathcal{D}(\mathbb{R}^n)$ implies convergence in $\mathcal{S}(\mathbb{R}^n)$, and $u_{|\mathcal{D}(\mathbb{R}^n)} \in \mathcal{D}'(\mathbb{R}^n)$ determines uniquely $u \in \mathcal{S}'(\mathbb{R}^n)$. Then a distribution $v \in \mathcal{D}'(\mathbb{R}^n)$ is the restriction of an element $u \in \mathcal{S}'(\mathbb{R}^n)$ if and only if there exist $N \in \mathbb{N}$ and a constant $C_N>0$ such that

1. $\displaystyle |u(\varphi)| \leq C_N q_N(\varphi)=C_N \sup_{x \in \mathbb{R}^n; |\alpha| \leq N} (1+|x|^2)^N |D^\alpha \varphi(x)|$ for all $\varphi \in \mathcal{D}(\mathbb{R}^n)$

where $q_N(\varphi)$ are seminorm that make $\mathcal{S}(\mathbb{R}^n)$ a Fréchet space. The elements of $\mathcal{S}'(\mathbb{R}^n)$ or their restriction to $\mathcal{D}(\mathbb{R}^n)$ are called tempered distributions. On $\mathcal{S}'(\mathbb{R}^n)$ we consider the topology $\sigma(\mathcal{S}'(\mathbb{R}^n), \mathcal{S}(\mathbb{R}^n))$, so that $u_k \rightarrow u$ in $\mathcal{S}'(\mathbb{R}^n)$ means that

2. $\displaystyle \langle \varphi, u_k \rangle\rightarrow \langle \varphi, u \rangle$ for all $\varphi \in \mathcal{S}(\mathbb{R}^n)$.

In reference to the comments, that is, of the various topologies. It is known that the Schwartz class $\mathcal{S}(\mathbb{R}^n)$ is a Fréchet space and also that the space of test functions $\mathcal{D}(\mathbb{R}^n)$ is dense in $\mathcal{S}(\mathbb{R}^n)$. In particular $\mathcal{D}_K(\mathbb{R}^n) \hookrightarrow \mathcal{S}(\mathbb{R}^n) \hookrightarrow L^1(\mathbb{R}^n)$ are continuous inclusions. Let $\mathcal{S}'(\mathbb{R}^n)$ the topological dual space $\mathcal{S}(\mathbb{R}^n)$. Now we have that the mapping $u \in \mathcal{S}'(\mathbb{R}^n) \longmapsto v=u_{|\mathcal{D}(\mathbb{R}^n)} \in \mathcal{D}'(\mathbb{R}^n)$ is linear and one-to-one because convergence in $\mathcal{D}(\mathbb{R}^n)$ implies convergence in $\mathcal{S}(\mathbb{R}^n)$, and $u_{|\mathcal{D}(\mathbb{R}^n)} \in \mathcal{D}'(\mathbb{R}^n)$ determines uniquely $u \in \mathcal{S}'(\mathbb{R}^n)$. Then a distribution $v \in \mathcal{D}'(\mathbb{R}^n)$ is the restriction of an element $u \in \mathcal{S}'(\mathbb{R}^n)$ if and only if there exist $N \in \mathbb{N}$ and a constant $C_N>0$ such that

1. $\displaystyle |u(\varphi)| \leq C_N q_N(\varphi)=C_N \sup_{x \in \mathbb{R}^n; |\alpha| \leq N} (1+|x|^2)^N |D^\alpha \varphi(x)|$ for all $\varphi \in \mathcal{D}(\mathbb{R}^n)$

where $q_N(\varphi)$ are seminorm that make $\mathcal{S}(\mathbb{R}^n)$ a Fréchet space. The elements of $\mathcal{S}'(\mathbb{R}^n)$ or their restriction to $\mathcal{D}(\mathbb{R}^n)$ are called tempered distributions. On $\mathcal{S}'(\mathbb{R}^n)$ we consider the topology $\sigma(\mathcal{S}'(\mathbb{R}^n), \mathcal{S}(\mathbb{R}^n))$, so that $u_k \rightarrow u$ in $\mathcal{S}'(\mathbb{R}^n)$ means that

2. $\displaystyle \langle \varphi, u_k \rangle\rightarrow \langle \varphi, u \rangle$ for all $\varphi \in \mathcal{S}(\mathbb{R}^n)$.

In other words, according to this definition of tempered distributions, the limit in (i) applies in any case: if $\psi \in \mathcal{D}(\mathbb{R}^n)$ or if $\psi \in \mathcal{S}(\mathbb{R}^n)$, because they are precisely restrictions of distributions.

I think this should formalize the last part of the proof.