- #1
SMA_01
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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.