In the text TAOCP
A real number is a quantity x that has a decimal expansion $$x = n + 0.d_1d_2d_3...$$ where n is an integer, each $d_i$ is a digit between 0 and 9, and the sequence of digits doesn't end with infinitely many 9s
Why is such a case ? Isn't all the numbers between 0 and 1 are all real ? Then the numbers which ends with infinitely many 9s should be real too, Isn't it ?
Yes, they are real. But note that $1=0.999999\ldots$, that $1.23=1.22999999\ldots$ and so on. What happens is that every real number has one and only one decimal expansion except those numbers which can be written as $\frac a{10^n}$ with $a\in\mathbb{Z}\setminus\{0\}$ and $n\in\mathbb Z$. These ones have two decimal expansions (as in my two examples above). A natural option then consists in not to consider the decimal expansions which end with infinitely many $9$'s.