Jtdylktuy

1 $begingroup$ I'm self-learning real analysis, and I am trying to understand a part in the proof for the following theorem in Mathematical Analysis by Apostol: Let $xgeq 0$ . Then for every integer $n geq 1$ , there exists a finite decimal $r_{n} = a_{0}.a_{1}a_{2}cdots a_{n}$ such that $r_{n} leq x < r_{n} + frac{1}{10^{n}}$ . Here is the given proof: let S be the set of all nonnegative integers that are less than or equal to $x$ . Since $0 in S$ , $S$ is not empty. Since $x$ is an upper bound of $S$ , $S$ has a supremum $a_0=sup(S)$ . Since $a_0le x$ , $a_0 in S$ , and so $a_0$ is a nonnegative integer. Then $a_0=lfloor x rfloor$ is the greatest integer in $x$ . Then $a_0le x < a_0 + 1$ . Let $a_1=lfloor 10x-10a_0rfloor$ be the greatest integer in $10x-10a_0$ . Then $0le 1