Theorem (Continuous Martingale with finite variation is constant)

Let $(X_{t})_{t\ge 0}$ be a continuous $(\mathcal{F}_{t})_{t\ge 0}$-martingale on $(\Omega,\mathcal{F},P)$. Assume that $\forall\omega\in\Omega: t\mapsto X_{t}(\omega)$ has finite variation on $\mathbb{R}^{+}$. Then $$\forall t\ge 0:X_{t}=X_{0}\text{ a.s.}$$