Triangle Inequality

Real Numbers
Let $$x, y \in \mathbb{R}$$ be real numbers.

Let $$\left|{x}\right|$$ be the absolute value of $$x$$.

Then:
 * $$\left|{x + y}\right| \le \left|{x}\right| + \left|{y}\right|$$;
 * $$\left|{x - y}\right| \ge \left|{x}\right| - \left|{y}\right|$$.

Complex Numbers
Let $$z_1, z_2 \in \mathbb{C}$$ be complex numbers.

Let $$\left|{z}\right|$$ be the modulus of $$z$$.

Then:
 * $$\left|{z_1 + z_2}\right| \le \left|{z_1}\right| + \left|{z_2}\right|$$;
 * $$\left|{z_1 - z_2}\right| \ge \left|{z_1}\right| - \left|{z_2}\right|$$.

Proof for Real Numbers
$$ $$ $$ $$ $$ $$

Then by Order of Squares in Totally Ordered Ring, $$\left|{x + y}\right| \le \left|{x}\right| + \left|{y}\right|$$.

From there we see $$\left|{x + y}\right| - \left|{y}\right| \le \left|{x}\right|$$.

Substitute $$z = x + y \Longrightarrow x = z - y$$ and so:

$$\left|{z}\right| - \left|{y}\right| \le \left|{z - y}\right|$$.

Renaming variables as appropriate gives $$\left|{x - y}\right| \ge \left|{x}\right| - \left|{y}\right|$$.