- #1

SMA_01

- 218

- 0

## Homework Statement

Prove llxl-lyll≤lx-yl

(The triangle inequality: la+bl≤lal+lbl)

## The Attempt at a Solution

For the first part, I assumed lxl≥lyl:

lxl=l(x-y)+yl

Then, by Triangle Inequality

l(x+y)+yl≤l(x-y)l+lyl

So,

lxl≤l(x-y)l+lyl

Subtract lyl from both sides to get:

lxl-lyl≤l(x-y)l.

I'm not sure where to go from here. For, llxl-lyll≤lx-yl, don't I need to prove -l(x-y)l≤lxl-lyl≤l(x-y)l? How would I finish the proof?

Thank you.