Infinitely many rational numbers between two different real numbers how to show that between any two distinct real numbers there are infinitely many rational numbers?  do we need to use the fact that a real number is an equivalent class of cauchy sequence of rational numbers?
 A: If you know that between any two real numbers there exists at least one rational number, you can conclude the result from there. Take $x,y\in\mathbb{R}$ so that $x\neq y$. Take a rational number $q_{1}\in (x,y)$. Since $q_{1}\neq x$, by similar reasoning we find a rational number $q_{2}\in (x,q_{1})$. In this fashion we obtain inductively for each $n\in\mathbb{N}$ a rational number $q_{n+1}\in (x,q_{n})$. In particular, $q_{k}\in (x,y)$ for all $n\in\mathbb{N}$, so between $x$ and $y$ there are infinitely many rational numbers.
To show that for any $x,y\in\mathbb{R}$ so that $x\neq y$ we find a rational number $q\in (x,y)$, we can use the fact that each real number is an equivalent class of Cauchy sequences.
A: Hint: If you can show there is one $q_1\in (a,b)$ where $a\neq b\in \mathbb{R}$, then you can show there are two: say a $q_2\in (a, q_1)\subset (a,b)$.  Then, you can find a third $q_3\in (a, q_2)$...
A: Let $a,b \in \mathbb{R}$, $a \not =b$. Without loss of generality suppose that $a<b$, then there exists a positive $c$ such that $b-a=c$. Now there exists a rational number $n> \frac{1}{c}$. Then $nc>1$ which implies that $n(b-a)>1$. Thus $nb-na>1$ so there exists an integer $r$ such that $na<r<nb$, hence $a< \frac{r}{n}<b$. Therefore there exists a rational number between any two real numbers. So there is a rational number $x$ such that $a<x<\frac{a+b}{2}$. Likewise there exists a rational number $x_1$ such that $\frac{a+b}{2}<x_1<b$ then we have that $a<x<x_1<b$. There exists another rational number $x_2$ such that $x_1<x_2<\frac{x_1+b}{2}$. Then there exists a rational number $x_3$ such that $\frac{x_1+b}{2}<x_3<b$. Again there exists another rational number $x_4$ such that $\frac{x_3+b}{2}<x_4<b$. Continuing in this fashion infintiely many times will result in an infinite number of rational numbers such that $a<x<x_1<x_2<x_3<x_4<\cdots<b$. 
My approach looking for suggestions on improvement. 
