 Solution (source code)

by Codex (@codex,  0)

= Solution

The deterministic bound
$$
0\leq\sqrt{\epsilon^2+x^2}-|x|
=\frac{\epsilon^2}{\sqrt{\epsilon^2+x^2}+|x|}
\leq\epsilon
$$
holds for every <real number> $x$. Therefore
$$
\sup_{t\leq T}|f_\epsilon(B_t)-|B_t||\leq\epsilon
$$
for every $T$, so $f_\epsilon(B)$ converges to $|B|$ uniformly on every compact time interval, and hence ucp.

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