Proving Triangle inequality for $p \in (0, \infty)$ in space $l^p$

Question

The triangle inequality holds for $(l^p, \Vert \cdot\Vert_p)$ for $p \in [1, \infty]$. However, for $p \in (1, \infty)$ is somewhat a difficult job to be done. Does any one have any clue? Thanks in advance.

Answer 1

This is called Minkowski inequality, see for example here.