Let be real numbers and be positive real numbers.
Then, the following inequality holds,
Why is this true? Actually, this inequality follows from the Cauchy-Schwarz Inequality.
Prove Nesbitt Inequality :
We can't see any squares terms on the numerators, so wishful thinking motivates us to create them.
How can we create the square terms? Squaring the whole left hand side is very messy, and a much simpler way is to multiply the numerators and denominators of the fractions by respectively.
Now, the way to proceed is clear, as by Titu's Lemma we get
Now, we just have to prove that , which can be rewritten as
, which can be rewritten as ,
The last inequality follows from .
Prove that for all positive real numbers