In this post, we show that if and are real numbers and is less than , then their arithmetic mean is greater than . That is, implies
Proof 1: Direct Proof
We know that .
Dividing both sides by 2, we have
Adding to both sides results to
Proof 2: Indirect Proof
Let us assume that for , .
Multiplying both sides by give us .
Subtracting from both sides results to .
This is in contradiction with the assumption above.
This means that the contradiction is false.