Are the elements of the set of adapted, almost surely increasing processes with $A_0=0$ finite variation processes?

Question

$\mathbb{A}_c=\{A_t \mid A \text{ adapted ,a.s continuous and increasing process,} A_0=0\}$

$\mathbb{V}_C=\{V_t \mid V \text{ adapted ,a.s continuous and finite variation process,} V_0=0\}$

Is $\mathbb{A}_c \subseteq \mathbb{V}_c $?