Intro to Real Analysis I am having trouble proving the following:

if $a < b$, then $a < {a+b\over2} < b$. 

I started with the Trichotomy Property and getting to where $a^2>0$, but then I do not know where to go from there. 
Any suggestions?

Given $a<b$,
$$a<b \Longleftrightarrow 2a<a+b \Longleftrightarrow a<\frac{a+b}{2}$$
$$a<b \Longleftrightarrow a+b<2b \Longleftrightarrow \frac{a+b}{2} < b$$
$$a<\frac{a+b}{2}<b$$
as desired.
