I have to prove the triangle inequality
$(|x_1 - z_1|^p + |x_2 - z_2|^p)^{1/p} \leq (|x_1 - y_1|^p + |x_2 - y_2|^p)^{1/p} + (|y_1 - z_1|^p + |y_2 - z_2|^p)^{1/p}$ for $p \geq 1$ on $\mathbb{R}^2$.
Two things are confusing me:
1) How we manipulate the expression, seeing as we can't use the standard triangle inequality $|x - z| \leq |x - y| + |y - z|$. I am used to using a $|x - z| = |x - z + y - y | < |x - y| + |y - z|$ method, however is this still valid given the powers of $p$?
2) Why this is an invalid statement for $0 < p < 1$
Sorry if these questions are a little basic, I'm just looking for a little guidance not a solution. Any help would be appreciated.
Thanks, Ash.