Let $M$ be a continuous martingale
Let $\varepsilon>0$. Note that $$\tau:=\inf\left\{t\in [0,T]:\left|M_t\right|\ge\varepsilon\right\}$$ is a stopping time with $$\left|M^\tau\right|\le\varepsilon\;.\tag1$$
Why is the stopped process $M^{\tau}$ bounded by epsilon. $\tau$ is the first time $|M_t| \geq \varepsilon$, so the stopped process cannot be bounded by this constant. What do I mix up?