Proof about real numbers

Question is from Apostol's Vol. 1 One-variable calculus with introduction to linear algebra textbook.

Page 28. Exercise 1. If $x$ and $y$ are arbitrary real numbers with $x

Any hints on how to approach the problem would be appreciated, other exercises seem to be similar so if I solve this I should be able to solve others as well. Thank you in advance.

Answer

Consider $z=\frac{x+y}{2}$. Then show that $x \lt z \lt y$.