According to several texts and professors, there exists a rational number between any two real numbers. But suppose you had two real numbers which had the same digits in the same places up to some place, where they differed by one digit – say the smaller one had a $4$ and the bigger one had a $5$. The smaller number's digits following the four are all $9$s, and the bigger number's digits after the five are all $0$s. For example:
$$ \beta=1.235\overline{0} \\ \alpha=1.234\overline{9}$$
What rational number is between $\alpha$ and $\beta$?
Best Answer
Have you seen the fact that $$0.99999\bar{9}=1.$$ These two numbers are literally equal. Take a look at this question to see more in depth answers regarding this fact: Is it true that $0.999999999\ldots = 1$?