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.
Lemma.
Given
Proof.
Observe that
with a similar equality for
We can now prove the famous arithmetic-mean/geometric-mean inequality.
Arithmetic-mean/Geometric-mean Inequality.
Let
Proof.
By induction on
Assume inequality
, so we would be done. Otherwise
we may assume (by re-ordering the
Thus
and by the induction hypothesis
giving
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.
Suppose
Proof.
With
so (1+