This is roughly exercise V.23 of Reed, Simon - Methods of Modern Mathematical Physics, vol. 1.
Let $\psi$ be a $\mathcal{C}^\infty$ function such that for every multi-index $\alpha$ there exist $c_\alpha> 0$ constant and a $N_\alpha$ such that
$$ \|\partial^\alpha \psi(x)\| \leq c_\alpha(1 + \|x\|^2)^{N_\alpha} $$
for every $x \in \mathbb{R}^d$.
(i) If $\varphi \in \mathcal{D}(\mathbb{R}^d)$, then $\varphi\psi \in \mathscr{S}(\mathbb{R}^d)$.
This is clear from Leibniz' Rule. What I'm having trouble to show is parts (ii) and (iii):
(ii) If $h$ is a measurable function such that $h\psi \in \mathscr{S}$ for all $\psi \in \mathscr{S}$, then $h$ has to be smooth.
(iii) Given $u \in \mathscr{S}'$, putting $(\psi u)(\varphi) = u(\psi\varphi)$ (as expected) for every $\varphi\in\mathscr{S}$, then $\psi u$ is a tempered distribution, i.e., $\psi u \in \mathscr{S}'$.
I'm actually not sure on how (ii) helps me at all on showing (iii) and how to show (ii).
Are you sure about the statement (i)? As stated, it is trivial because for $\varphi\in\mathscr D$ and smooth $\psi$ one has $\varphi\psi\in\mathscr D \subseteq \mathscr S$. Perhaps you mean $\varphi\psi\in\mathscr S$ for all $\varphi\in \mathscr S$? Then the map $\varphi\mapsto \varphi\psi$ is continuous on $\mathscr S$ by the closed graph theorem and $\psi u$ is continuous as a composition. This solves (3).
I don't see, why (ii) should be useful. For the proof of (ii) just consider test functions $\psi\in\mathscr D$ which are constant on a neighbourhood of the point where you want to show smoothness.