prove that $a<\frac{a+b}{2}<b$

Question

I was looking on a problem in which we need to find rational numbers between 2 numbers. In the solution it was told that to find rational numbers between a and b we need to find $\frac{a+b}{2}$ i.e., $a<\frac{a+b}{2}<b$. But I wanted to know if there any prove that satisfies $a<\frac{a+b}{2}<b$. If yes, then what is the proof?

Answer 1

If $a<b$, then $\frac a2<\frac b2$.

Now, take the inequality $\frac a2<\frac b2$ and try to add some number to both sides. Try to add such a number that one side becomes $a$.

Answer 2

Suppose $a \leq b$. Then $(a+a)\; \leq \;(a + b)\; \leq (b+b)$. Dividing by two, we find the desired result: $$a \leq \frac{a+b}{2} \leq b.$$