A previous page showed from the monotone convergence theorem
that in the reals, all positive
Recall that given numbers
We will prove this, and the proof will be by induction.
The following lemma will be the base case of the induction.
with a similar equality for
We can now prove the famous arithmetic-mean/geometric-mean inequality.
By induction on
inequality, so we would be done. Otherwise
we may assume (by re-ordering the
and by the induction hypothesis
We give as a corollary a useful inequality, reminiscent of Bernoulli's inequality which we saw earlier, but the inequality sign goes the other way round and the result is only valid for exponents at most one.
The Exponential Inequality.