= Solution
Each $A^\epsilon$ is continuous, adapted, and increasing. From ucp convergence one can choose a <convergent subsequence>[subsequence] that converges uniformly almost surely on every compact interval. Its limit $A$ is therefore also continuous and increasing, hence a <finite-variation process>. The <stochastic integral> $M_t=\int_0^t\operatorname{sgn}(B_s)dB_s$ is a <continuous local martingale>, and part e gives the <semimartingale decomposition>
$$
|B_t|=M_t+A_t.
$$
Consequently $|B|$ is a semimartingale. In fact, comparison with the <Tanaka formula> identifies $A_t$ as the <local time of a semimartingale> $L_t^0(B)$.
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