= Solution
Choose stopping times $T_m\uparrow\infty$ such that $X^{T_m}$ is a bounded martingale. Part a constructs $[X^{T_m}]$. Uniqueness in the identity
$$
(X^{T_m})^2-[X^{T_m}]\text{ is a local martingale}
$$
shows consistency on overlapping stopped intervals, so define $[X]_{t\wedge T_m}=[X^{T_m}]_t$. The stopped dyadic sums converge uniformly on every compact interval in probability, and
$$
X^2-[X]
$$
is a local martingale. This localization constructs the <quadratic variation> of every continuous <local martingale>.
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