Lipschitz implies uniformly continuous

Let \((X, d_X)\) and \((Y, d_Y)\) be metric spaces and let \(f\colon X\to Y\) be Lipschitz: there exists \(C > 0\) such that \(d_Y(f(x_1), f(x_2)) \leq C\, d_X(x_1, x_2)\) for all \(x_1, x_2 \in X\). Then \(f\) is uniformly continuous.