Uniform limits of continuous functions are continuous

Let \(E\) be a metric space, and let \(f_n\colon E\to\RR\) for \(n\geq1\) and \(f\colon E\to\RR\) be functions. If each \(f_n\) is continuous on \(E\) and \(f_n\to f\) uniformly on \(E\), then \(f\) is continuous on \(E\). [@Rud76].