- #1
scharl4
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Homework Statement
Using the generalized triangle inequality, prove |d(x,y) - d(z,w)| ≤ d(x,z) + d(y,w)
Homework Equations
d(x,y) is a metric
triangle inequality: d(x,y) ≤ d(x,z) + d(z,y)
The Attempt at a Solution
I know that this needs to be proved with cases: a) d(x,y) - d(z,w) ≥ 0, and b) d(x,y) - d(z,w) < 0. I know that for case a), we have |d(x,y) - d(z,w)| = d(x,y) - d(z,w). The part I am stuck at is that by applying the generalized triangle inequality to this expression, we are supposed to get d(x,y) - d(z,w) ≤ d(x,z) + d(w,y). I just do not understand why. Can anyone help please?