= Solution
For the <integrable terminal bracket criterion>, assume $M_0=0$, or more generally $M_0\in L^2$. This initial-value assumption is necessary if it is not already part of the notation: an integrable but non-square-integrable random constant process has zero <quadratic variation> and still fails square-integrability.
For the zero-starting case, choose a <localizing sequence> $\tau_n$ bounding both $|M|$ and its <quadratic variation>. The stopped processes are true square-integrable <martingales>, and the <Itô isometry> gives
$$
\mathbb E[M_{t\wedge\tau_n}^2]=\mathbb E[M]_{t\wedge\tau_n}\le C:=\mathbb E[M]_\infty<\infty.
$$
For each fixed time these variables are <uniformly integrable> by their bounded second moments. Almost sure convergence as $n\to\infty$ is therefore convergence in $L^1$. Passing to the limit in the stopped conditional-expectation identity gives $\mathbb E[M_t\mid\mathcal F_s]=M_s$: the <local martingale> is a true <martingale>. The <Fatou lemma> also gives
$$
\boxed{\sup_{t\ge0}\mathbb EM_t^2\le\mathbb E[M]_\infty<\infty},
$$
so $M$ belongs to the global $\mathcal M_c^2$ space. For general square-integrable $M_0$, apply this argument to $M-M_0$ and use $|M_t|^2\le2|M_0|^2+2|M_t-M_0|^2$.
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