Notes - Analysis I MT22, Triangle inequality
Triangle inequality
When proving the triangle inequality for the reals, i.e.
If $x, y \in \mathbb{R}$ then $ \vert x+y \vert \le \vert x \vert + \vert y \vert $.
What two ingredients do you start with?
If $x, y \in \mathbb{R}$ then $ \vert x+y \vert \le \vert x \vert + \vert y \vert $.
- $- \vert x \vert \le x \le \vert x \vert $
- $- \vert y \vert \le y \le \vert y \vert $
Reverse triangle inequality
Can you state the reverse triangle inequality for the reals?
\[\big||x| - |y|\big| \le |x-y|\]
The reverse triangle inequality states
If $x, y \in \mathbb{R}$ then $\big \vert \vert x \vert - \vert y \vert \big \vert \le \vert x - y \vert $.
What two inequalities do you break this down into?
If $x, y \in \mathbb{R}$ then $\big \vert \vert x \vert - \vert y \vert \big \vert \le \vert x - y \vert $.
- $- \vert x-y \vert \le \vert x \vert - \vert y \vert $
- $ \vert x \vert - \vert y \vert \le \vert x-y \vert $
What’s the general strategy for proving things like
- $- \vert x-y \vert \le \vert x \vert - \vert y \vert $, or
- $ \vert x \vert - \vert y \vert \le \vert x-y \vert $
that look roughly like the triangle inequality?
Make substitutions for $a, b$ in $ \vert a + b \vert \le \vert a \vert + \vert b \vert $