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
which is the triangle inequality for the case n= k+1.
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Author: Michael O'Neill