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