Triangle Inequality/Real Numbers/Proof 5

Proof
It suffices to prove that:


 * $\\ \size{x}+\size{y} \ge x+y$


 * and


 * $ \size{x}+\size{y} \ge -(x+y)$.

By absolute value properties, we have that:


 * $\\ x\le\size{x}$


 * and


 * $y\le\size{y}$

Then:
 * $x + y \le \size{x} + \size{y}$

We also have that:


 * $\\-x\le\size{-x}=\size{x} \Rightarrow -x\le\size{x}$


 * and


 * $\\-y\le\size{-y}=\size{y} \Rightarrow -y\le\size{y}$

Then:


 * $\\-(x+y) = -x + (-y) \le \size{x} + \size{y}$

So,


 * $-(x+y) \le \size{x} + \size{y}$

As we have proof of the two sufficient statements, we have that:


 * $\\ \size{x+y} \le \size{x} + \size{y}$