= Solution
For every finite collection of positive times, $(B_{t_1},\ldots,B_{t_m})$ is a linear transformation of the corresponding values of $W$, so $B$ is a centered <Gaussian process>. If $0<s\leq t$, then
$$
\mathbb E[B_sB_t]
=st\,\mathbb E[W_{1/s}W_{1/t}]
=st\min\!\left(\frac1s,\frac1t\right)
=s
=\min(s,t).
$$
Thus $B$ has the covariance of <Brownian motion>.
Its paths are continuous for $t>0$. As $t\downarrow0$, put $u=1/t$. The given almost-sure limit gives
$$
B_t=tW_{1/t}=\frac{W_u}{u}\longrightarrow0=B_0.
$$
The <Gaussian-process characterization of Brownian motion> now proves the <time inversion of Brownian motion>:
$$
\boxed{(B_t)_{t\geq0}\text{ is a standard Brownian motion}.}
$$
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