We denote $$M_{\mathrm{loc}} ^{0}:=\{\text{set of cont. local mart. starting in 0}\}$$ $$A:=\{\text{set of cont. processes of fin. variation}\}$$
To prove that their intersection is $\{0\}$ we first prove that any bounded process in their intersection is zero, and than choosing a suitable sequence of stopping times extend to any process.
Now let $X\in M_{\mathrm{loc}}^0 \cap A$ and bounded. Fix $\epsilon \gt 0$ and set $$T_0=0, T_{n+1}:=\inf\left\{t\gt T_n: \left|X_t-X_{T_n}\right|\gt\epsilon\right\}$$ Since $X\in A$ than the stopping times go to $\infty$ (a.s.). Now if I show that the process $\left(X_{T_n}\right)_{n\ge0}$ is a martingale than I'm able to show the statement by a variation argument, but I'm unable to show that this is actually a martingale. Do you have any ideas?