TITLE: Infinite number of rationals between any two reals.
QUESTION [15 upvotes]: Let $a$ and $b$ be reals with $a<b$. Show that there are infinitely many rationals $x$ such that $a<x<b$.
My plan of action was to assume that $x$ is the smallest such rational and find another rational in the interval $(a, x)$, but I am struggling to make it work. A hint will be much preferred to a full solution.

REPLY [5 votes]: If you already know (or can prove) that there is at least one rational 
between any two real numbers, then you can do this for $a < b$:
There is a rational number $x$ such that $a < x < \frac{a+b}{2}.$
There is a rational number $y$ such that $\frac{a+b}{2} < y < b.$
Now $a < x < y < b,$ with $x$ and $y$ rational.