Easy Way to See Reverse Triangle Inequality

To Show

\[\| |x| - |y| \| \leq |x - y|\]

Easy Way To See It

Remember that the absolute value/norm of the difference of 2 vectors is the distance between them. Rewrite \(\| |x| - |y| \|\) as \(d(|x|, |y|)\) and \(|x-y|\) as \(d(x,y)\).

Now we want to show:

\[d(|x|, |y|) \leq d(x, y)\]

But this is obvious. Making both terms the same sign can only put them on the same side of the number line, which can never increase the distance between them.

And this is why I prefer \(d(x, y)\) to \(\|x - y\|\).

Related Posts

Just because 2 things are dual, doesn't mean they're just opposites

Boolean Algebra, Arithmetic POV

discontinuous linear functions

Continuous vs Bounded

Minimal Surfaces

November 2, 2023

NTK reparametrization

Kate from Vancouver, please email me

ChatGPT Session: Emotions, Etymology, Hyperfiniteness

Some ChatGPT Sessions