Let \(\{f_n:[0,1]\to\mathbb R\}_{n=1}^{\infty}\) be continuous functions that converge uniformly to \(0\). Show that \[ \int_0^1 f_n(x)\,dx\longrightarrow0. \]
Solution
Uniform convergence to zero gives \[ \|f_n\|_{\infty}=\sup_{x\in[0,1]}|f_n(x)|\longrightarrow0. \] Therefore \[ \left|\int_0^1f_n(x)\,dx\right| \leq\int_0^1|f_n(x)|\,dx \leq\|f_n\|_{\infty}\longrightarrow0. \]