[Math] Lipschitz continuity implies continuity

continuitylipschitz-functionsreal-analysis

Say $$\vert f(x)-f(y)\vert \le L\vert x-y\vert$$. How to prove the following: $$\forall \lim_{n \to \infty} x_n = x_0 \wedge x_0\in \Bbb{R}$$: $$\lim_{n \to \infty}f(x_n) = f(x_0)$$?

In other words: how to prove that Lipschitz-continuity implies regular continuity(without using differentiation or similar methods)?

$$|f(x_n)-f(x_0)|\le L|x_n-x_0|$$
$$\implies$$
$$f(x_0)-L|x_n-x_0|\le f(x_n)\le f(x_0)+L|x_n-x_0|$$