Divergent logarithmic clock for a positive local martingale tending to zero ($M_t\to 0⟹⟨X{⟩}_{\infty }=\infty$)

Write a positive continuous local martingale as $M=\exp (X-⟨X⟩/2)$ using its stochastic logarithm. If $M_t\to 0$, its logarithmic bracket must diverge. Otherwise the finite-bracket convergence lemma makes $X$ converge finitely and the exponential has a positive limit.