[proofplan] We apply the standard estimate $|\int g| \leq \int |g|$ for complex-valued Riemann integrals to the parametrisation of the contour [integral](/page/Integral), then bound $|f(\gamma(t))|$ by $M$ and factor out the length integral $\int_a^b |\gamma'(t)| \, d\mathcal{L}^1(t) = \ell(\gamma)$. [/proofplan] [step:Bound the contour integral by parametrising and applying the modulus estimate] By definition of the contour [integral](/page/Integral): \begin{align*} \left| \int_\gamma f(z) \, dz \right| = \left| \int_a^b f(\gamma(t)) \gamma'(t) \, d\mathcal{L}^1(t) \right|. \end{align*} We apply the standard estimate for complex-valued Riemann integrals: for any integrable $g: [a,b] \to \mathbb{C}$, $|\int_a^b g(t) \, d\mathcal{L}^1(t)| \leq \int_a^b |g(t)| \, d\mathcal{L}^1(t)$. (This is proved by writing $\int g = re^{i\theta}$ and observing $r = \operatorname{Re}(e^{-i\theta} \int g) = \int \operatorname{Re}(e^{-i\theta} g) \leq \int |g|$.) Therefore: \begin{align*} \left| \int_a^b f(\gamma(t)) \gamma'(t) \, d\mathcal{L}^1(t) \right| \leq \int_a^b |f(\gamma(t))| \cdot |\gamma'(t)| \, d\mathcal{L}^1(t). \end{align*} Since $|f(\gamma(t))| \leq M$ for all $t \in [a,b]$: \begin{align*} \int_a^b |f(\gamma(t))| \cdot |\gamma'(t)| \, d\mathcal{L}^1(t) \leq M \int_a^b |\gamma'(t)| \, d\mathcal{L}^1(t) = M \cdot \ell(\gamma). \end{align*} [/step]