Absolute value of a martingale

Question

Given a martingale $M(t)$, can I use Doob's inequality on $|M(t)|$ to achieve the following upper bond? $$P(\sup_{0\leq t\leq T}|M(t)|>\epsilon)=P(\sup_{0\leq t\leq T}|M(t)|^2>\epsilon^2)\leq\frac{E[|M(T)|^2]}{\epsilon^2}.$$ My reasoning is that since $M(t)$ is a martingale, both $|M(t)|$ and $|M(t)|^2$ are ones too.