Prove that if $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ are vectors in $R^{n}$, then: $(a) \quad\|\mathbf{u}+\mathbf{v}\| \leq\|\mathbf{u}\|+\|\mathbf{v}\| \quad$ [Triangle inequality for vectors] $(b) \quad d(\mathbf{u}, \mathbf{v}) \leq d(\mathbf{u}, \mathbf{w})+d(\mathbf{w}, \mathbf{v}) \quad$ [Triangle inequality for distances]