Let’s prove that without a calculator. If you haven’t seen this before, give it a try before you read any further.
Consider the function on the interval . This function shoots off to positive infinity as tends towards either endpoint of the domain. Let’s find its minimum value via an application of good clean differential calculus.
Differentiating (remember the quotient rule?) with respect to gives us that
Then, we have that when , so clearly the stationary point of this function occurs at . It is also easy to check that this is a minimum using your favourite high-school test.
Therefore, as the minimum value of occurs when , then if we choose any number which is not equal to we have
For instance, we can choose . Using the fact that , this gives us that
This inequality can then be manipulated to get that . The sharp reader will notice that any real number greater than 1 and not equal to can replace in the inequality and it shall still be true!