In this post, we show that if $latex a$ and $latex b$ are real numbers and $latex a$ is less than $latex b$, then their arithmetic mean is greater than $latex a$. That is, $latex a < b$ implies

$latex a < \displaystyle \frac{a + b}{2}$.

Proof 1: Direct Proof

We know that $latex a < b$.

Dividing both sides by 2, we have

$latex \displaystyle \frac{a}{2} < \displaystyle \frac{b}{2}$.

Adding $latex \frac{a}{2}$ to both sides results to  Continue reading →