= Solution
Strict positivity lets us define the <continuous local martingale>
$$
\boxed{X_t=\int_0^t\frac{dM_s}{M_s}.}
$$
The integrand is locally bounded because a positive continuous path has positive minimum on every compact time interval. The <Itô formula> for $\log M$ gives
$$
\log M_t=X_t-\frac12\langle X\rangle_t,\qquad\boxed{M_t=\exp\left(X_t-\frac12\langle X\rangle_t\right).}
$$
This is the <stochastic exponential> representation of a positive <continuous local martingale>.
If $\langle X\rangle_\infty$ were finite on an event of positive probability, the <finite-bracket convergence lemma> proved in Question 2(a) would make $X_t$ converge to a finite limit there. The exponential would then have a strictly positive limit, contradicting the assumed $M_t\to0$. Therefore
$$
\boxed{\langle X\rangle_\infty=\infty\quad\text{almost surely}.}
$$
This is the <divergent logarithmic clock for a positive local martingale tending to zero>.
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