Help with math problem?

Question

Please help me show that: abs value[ (w^2 + x^2) - (y^2 + z^2) ] < C * sqrrt[ (w-y)^2 + (x-z)^2 ] for some positive constant C, and all real numbers: w, x, y, z

Answer 1

It isn't true.

let w = x = n
let y = z = n-1
In other words, pick a number for w and x to equal then choose y and z to be a number 1 less than w and x.

This will always make the right hand side (RHS) a constant
C * sqrt[ ( n - (n-1) )^2 + (n - (n-1) )^2 ] =
C * sqrt[ (1^2 + 1^2) =
C * sqrt(2).

The left hand side (LHS) would be
abs [n^2 + n^2 - (n-1)^2 - (n-1)^2 ] =
abs [2n^2 - 2 * (n-1)^2] =
abs [2n^2 - 2 * (n^2 - 2n + 1)] =
abs[4n - 2 ]

If you pick n = 1, then the LHS = 2 so you can pick C to be a number greater than sqrt 2. Because c = sqrt(2) would give you 2 on the RHS.
But you can see that if you keep picking greater values of n, the LHS keeps getting greater but the RHS stays the same for a specific C. So, no matter what C you pick you can find an n that makes the inequality false.