## Simpel Quenstion (maybe) about stoppingtime

I have an adaptet, cadlag proces $M$ with $M(0)=0$. Then I have that the jumps $\Delta M \geq 0$ and a stoppingtime

$
\tau_{x}=inf \{ t \geq 0 : M(t) - \mu t $

But why is it that

$M( \tau_{x} )= \mu \tau_{x} + x$

on $(\tau_{x} < \infty)$?

And not $M( \tau_{x} ) < \mu \tau_{x} + x$

Can anyone explain this?