= Solution
For a <simple predictable process>
$$
H_s=\sum_{i=0}^{n-1}H_i\mathbf1_{(t_i,t_{i+1}]}(s),
\qquad H_i\in L^\infty(\mathcal F_{t_i}),
$$
define
$$
(H\mathbin\cdot M)_t
=\sum_{i=0}^{n-1}H_i
\bigl(M_{t\wedge t_{i+1}}-M_{t\wedge t_i}\bigr).
$$
Each summand is a bounded predictable multiple of a martingale increment, so conditional expectation proves that $H\mathbin\cdot M$ is a martingale. Orthogonality of disjoint martingale increments gives
$$
\mathbb E|(H\mathbin\cdot M)_\infty|^2
=\sum_i\mathbb E\!\left[
H_i^2(M_{t_{i+1}}-M_{t_i})^2\right]<\infty.
$$
It is therefore an $L^2$-bounded continuous martingale.
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