Show that if \(f_n\to f\) uniformly then \(\int_\gamma f_n\to \int_\gamma f\).
Solution
\[ \abs{ \int_\gamma f_n(z) \dz - \int_\gamma f(z) \dz } &= \abs{\int_\gamma f_n(z) - f(z) \dz} \\ &\leq \int_\gamma \abs{f_n - f} \abs{\dz} \\ &\leq \int_\gamma \norm{f_n - f}_{\infty, \gamma} \cdot \abs{\dz} \\ &= \eps \cdot \length(\gamma) \\ &\convergesto{n\to\infty} 0 .\]