\begin{align*} \forall x_1,x_2 \in \mathbb{R} \ni x_1 \lt x_2\\ F(x_2) & = \int_{-\infty}^{x_2} f(x) dx \\ & = \int_{-\infty}^{x_1} f(x) dx + \int_{x_1}^{x_2} f(x) dx\\ & \ge \int_{-\infty}^{x_1} f(x) dx\\ & = F(x_1) \end{align*}