= Solution
Take an increasing <localizing sequence> $\tau_n$ for $X$ and define the <stopping times>
$$
R_n=\inf\{u\geq0:|K_{u+1}|>n\},\qquad T_n=\tau_n\wedge R_n\wedge n.
$$
The <predictable process> condition $K_{u+1}\in\mathcal F_u$ makes $R_n$ a <stopping time>. For each fixed finite time range, the finitely many coefficients $K_1,\ldots,K_t$ are finite <almost surely>; hence $T_n\uparrow\infty$ <almost surely>.
The stopped <martingale transform> can be written
$$
M_{t\wedge T_n}=\sum_{s=1}^tK_s\mathbf1_{\{R_n\geq s\}}\mathbf1_{\{n\geq s\}}\bigl(X_{s\wedge\tau_n}-X_{(s-1)\wedge\tau_n}\bigr).
$$
Its coefficient is $\mathcal F_{s-1}$-measurable and bounded in absolute value by $n$. Each summand is consequently integrable with zero <conditional expectation> given $\mathcal F_{s-1}$, since $X^{\tau_n}$ is a true <martingale>. Thus $M^{T_n}$ is a true <martingale> and
$$
\boxed{M=K\mathbin\cdot X\ \text{is a local martingale}.}
$$
The original PDF fixes $M_0=0$; the TeX extraction omits this equality.
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