# Prove that PSK|a−b|\geq|a|−|b|ZSK.

Prove that $$\displaystyle{\left|{a}−{b}\right|}\geq{\left|{a}\right|}−{\left|{b}\right|}$$.

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Terry Ray
It is called the reverse triangle inequality.
$$\displaystyle{\left|{a}−{b}\right|}\geq{\left|{\left|{a}\right|}−\right|}{b}{\mid}{\mid}$$
$$\displaystyle{\left|{a}\right|}={\left|{a}-{b}+{b}\right|}\le{\left|{a}-{b}\right|}+{\left|{b}\right|}$$
$$\displaystyle{\left|{b}\right|}={\left|{b}-{a}+{a}\right|}\le{\left|{a}-{b}\right|}+{\left|{a}\right|}$$
Thus, we have:
$$\displaystyle-{\left|{a}-{b}\right|}\le{\left|{a}\right|}-{\left|{b}\right|}\le{\left|{a}-{b}\right|}$$
###### Not exactly what you’re looking for?
Tiefdruckot
The length of any side of a triangle is greater than the absolute difference of the lengths of the other two sides:
$$\displaystyle{\left|{\left|{a}\right|}−\right|}{b}{\left|\le\right|}{a}−{b}{\mid}$$
So,
$$\displaystyle{\left|{a}−{b}\right|}\geq{\left|{a}\right|}−{\left|{b}\right|}$$
$$\displaystyle{\left|{a}−{b}\right|}\geq{\left|{b}\right|}−{\left|{a}\right|}$$
That's it.