Martingale and Lenglart inequality

Question

Suppose that $M$ is a square-integrable Martingale with $M_0=0$. I want to apply the Lenglart inequality to $M$ concluding that $$ \mathbb{P}\left[\sup_{s\leq t}\left|M_s\right|\geq \varepsilon\right]\leq \frac{\eta}{\varepsilon}+\mathbb{P}\left[\left<M\right>_t\geq\eta\right] $$ In order to do so I need to prove that $M$ is $L$-dominated by $\left<M\right>$, i.e. $\mathbb{E}[\left|M_{\tau}\right|]\leq\mathbb{E}[\left<M\right>_{\tau}]$ for every bounded stopping time $\tau$. What I can prove is the following $$ \mathbb{E}[\left|M_t\right|]\leq \left(\mathbb{E}[M_t^2]\right)^{1/2} =\left(\mathbb{E}[M^{\prime}_t+\left<M\right>_t]\right)^{1/2}=\left(\mathbb{E}[\left<M\right>_t]\right)^{1/2},\quad(1) $$ where I have used the Jensen's inequality and the Doob-Meyer decomposition $M^2_t=M^{\prime}_t+\left<M\right>_t$, with $M^{\prime}_t$ martingale with $M^{\prime}_0=0$ and so $\mathbb{E}[M^{\prime}_t]=0$. Nevertheless I am not sure that $(1)$ is enough to use the Lenglart inequality.