#projections

True or false: projecting onto the unit circle $\{\mathbf{x} : \lVert \mathbf{x} \rVert_2 = 1\}$ (the curve, not the disc), $P(\mathbf{x}) = \mathbf{x} / \lVert \mathbf{x} \rVert_2$, never increases distances: $\lVert P(\mathbf{x}) - P(\mathbf{y}) \rVert \le \lVert \mathbf{x} - \mathbf{y} \rVert$ for all nonzero $\mathbf{x}, \mathbf{y}$.