Pseudonorm from royden book

Question

How to show the triangle in equality for the following norm $\|x\|=inf_{y\in Y}\|x-y\| $, such that $X$ is a normed linear space and $Y$ is a closed linear subspace of $X$.

Answer 1

Hint: Note that $$ \inf_{y \in Y} (\|x_1 - y\| + \|x_2-y\|) \leq \inf_{y \in Y} \|x_1 - y\| + \inf_{y \in Y} \|x_2 - y\| $$ since $$ \{\|x_1 - y\| + \|x_2-y\| : y \in Y\} \subset \{\|x_1 - y_1\| + \|x_2-y_2\| : y_1 \in Y \text{ and } y_2 \in Y\} $$