Consider a twice diferentiable function $F$ on $R$ with bounded first derivative $F'$ and a Brownian motion $W$. Show that $F(W_t)-\frac{1}{2} \int_{0}^{t} F'' (W_s)ds$ is a true martingale.

I tried do show it using this, but I only got confused and did not find any solution.

If $M$ is a local martingale with continues trajectories, then it is a true martingale and $E(M_t^2) < \infty$ for all $t\geq 0$ Or

If $M$ is a local martingale with continues trajectories, then it is a true martingale and $E([M]_t)<\infty$ for all t.