$f$ is a continuous function and $|f(z)| \leq M$. I need to prove $ \Big|\int_{\gamma} f(z) dz \Big| \leq M \ell (\gamma) $ where $\ell (\gamma) = \int_{0}^1 |\gamma'(t)| dt$.
Here is my attempt: $ \Big|\int_{\gamma} f(z) dz \Big|= \Big|\int_{0}^1 f(\gamma(t))\gamma'(t) dt\Big |\leq\int_{0}^1 \Big|f(\gamma(t))\Big |\Big |\gamma'(t)\Big | dt\leq \int_{0}^1 M\Big |\gamma'(t)\Big | dt =M\int_{0}^1 |\gamma'(t)| dt=M \ell (\gamma) $
Is this straightforward prove right? Thanks!