Let $B \in \text{Wiener}(0, 1)$ be a standard Wiener process. By definition
\begin{equation}
W_t
\overset{d}{=} \mu t + \sigma B_t
\end{equation}
Result
R5378: Distribution of the standard real Wiener process at a given point is gaussian shows that
\begin{equation}
B_t
\overset{d}{=} \text{Gaussian}(0, \sqrt{t})
\end{equation}
The claim follows. $\square$
