Does the mean of two positive numbers obey these basic inequalities?

Question

Suppose $a,b$ are positive real numbers. Does it follow that $$ a < \frac{a+b}{2} < b $$ provided $a < b$?

Answer 1

(Multiplying each expression in the inequalities shows that) the statement is equivalent to $$a + a < a + b < b + b,$$ and each of these inequalities follow from $a < b$.

Answer 2

Mark two different numbers a and b such that $a<b$ on the real line and then compute arithmetic mean of two and observe where it lies.

Also if you reduce the above inequality into two $(a+b)/2> a$ and $(a+b)/2 < b$ and simplify it you would get $a<b$ which means your result holds good.