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?
[Math] Infinitely many rational numbers between two different real numbers
real-analysis
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Best Answer
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.