Understanding Mathematics by Peter Alfeld, Department of Mathematics, University of Utah

A Proof of the Reverse Triangle Inequality

Let's suppose without loss of generality that ||x|| is no smaller than ||y||. (Otherwise we just interchange the roles of x and y.) Thus we have to show that

This follows directly from the triangle inequality itself if we write x as


and think of it as

x=(x-y) + y.

Taking norms and applying the triangle inequality gives

which implies (*).

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