# Proof for range of average

We can prove that average of two numbers $$a,b$$ where $$a will be between $$a$$ and $$b$$ as follows

$$a < b$$
$$a + a < a + b$$
a < $$\dfrac{a + b}{2 }$$

$$a < b$$
$$a + b < b + b$$
$$\dfrac{a + b}{2 }< b$$

Thus
$$a < \dfrac{a + b}{2 }< b$$

Similarly, is it true always and can be proved, that average of three numbers, $$a,b,c$$ where $$a will be between $$a$$ and $$b$$ if $$b-a$$ > $$c-b$$

From what you've already got, we have $$a $$\implies 2a
Also, we have $$b-a>c-b\implies a+c\lt 2b\implies a+b+c\lt 3b\implies \frac{a+b+c}{3}\lt b$$