### A small note on re-defining variables to prove inequalities

I just want to record my solution to the following problem, as it is different from the one given online.

For positive real numbers, prove that

This has a fairly straight forward solution using Cauchy-Schwarz inequality, which for some reason I did not think of.

The way that I solved it is that I re-defined the variables: let and . Then this is equivalent to proving that

This is easily seen to be a consequence of Jensen’s inequality, as is a convex function for positive .