Does $d(x,y)$ satisfy the triangle inequality? Let $d(x,y)=\sup_{j\in N}|\xi_j-\eta_j|$ where $x=(\xi_1,\xi_2,...)$ and $y=(\eta_1,\eta_2,...)$.

Does $d(x,y)$ satisfy the triangle inequality?
Yes, $d$ satisfies the triangle inequality. Reason: $|*|$ satisfies the triangle inequality. Now its your turn to give a proof.
Yes, $d$ satisfies the triangle inequality. Reason: $|*|$ satisfies the triangle inequality. Now its your turn to give a proof.