Triangle Inequality/Real Numbers/Proof 4

Proof
We do a case analysis.

$(3): \quad x \ge 0, y \le 0$
We have that $\size x = x$ and $\size y = -y$.

In this case we show:


 * $\size {x + y} \le \max \set {\size x, \size y}$

Let $\size x \le \size y$.

Then:

Let $\size x \ge \size y$.

Then:

We have $\max \set {a, b} \le a + b$ for positive real numbers $a$ and $b$.

The result follows by taking $a = \size x$ and $b = \size y$.

$(4): \quad x \le 0, y \ge 0$
Follows by symmetry from the case $(3)$.