$\begin{align*}\| (x_0, y_0) \| = \| (x_0, y_0) - (x, y) + (x, y) \| \leq \underbrace{\| (x_0, y_0) - (x, y) \|}_{= d((x_0, y_0), (x, y))} + \| (x, y) \| < \| (x_0, y_0) \| + \| (x, y) \|\end{align*}$