Let $C$ be a non-empty, convex and closed subset of $\mathbb{R}^d$. Show that for all $(x_1, x_2) \in \mathbb{R}^d \times \mathbb{R}^d$, we have
$$\left(\operatorname{proj}_C(x_1) - \operatorname{proj}_C(x_2)\right) \cdot \left(x_1 - x_2\right) \geqslant \left\|\operatorname{proj}_C(x_1) - \operatorname{proj}_C(x_2)\right\|^2$$
and deduce that $\operatorname{proj}_C$ is continuous.