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

which equals

.

**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.

Therefore, .