The triangle inequality for n complex numbers.

We know the inequality when n=1 and when n=2 by the last exercise. We will show that the truth of the inequality for n=k implies it for n=k+1 when k is any integer. That will finish the proof. This is an example of proof by induction.

By the triangle inequality (in the simplest case n=2),

So the inductive hypothesis that

implies

which is the triangle inequality for the case n= k+1.

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