Prove that every real number x in [0,1] has a decimal expansion.

1. Mar 21, 2013

robertjordan

1. The problem statement, all variables and given/known data

Prove that every real number x in [0,1] has a decimal expansion.

2. Relevant equations

Let $x\in{[0,1]}$, then the decimal expansion for x is an infinite sequence $(k_{i})^{\infty}_{i=1}$ such that for all i, $k_i$ is an integer between 0 and 9 and such that $x\in\left[\frac{k_1}{10}+\frac{k_2}{10^2}+\cdots +\frac{k_n}{10^n},\frac{k_1}{10} +\frac{k_2}{10^2}+ \cdots +\frac{k_{n}+1}{10^n}\right]$.

We call that interval above $I_{k_1,k_2,\ldots,k_n}$

3. The attempt at a solution

Assume BWOC that there exists a real number $t\in[0,1]$ with no decimal expansion. That means there exists an $N\in{\mathbb{Z}}$ such that for all sequences $(k_{i})^{N}_{i=1}$, $t{\notin}\left[\frac{k_1}{10}+\frac{k_2}{10^2}+\cdots +\frac{k_N}{10^N},\frac{k_1}{10}+ \frac{k_2}{10^2}+ \cdots +\frac{k_{N}+1}{10^N}\right]$.

But $I_{0,0,\ldots,0}\cup I_{0,0,\ldots,0,1} \cup I_{0,0,\ldots,0,2} \cup \cdots \cup I_{9,9,9,\ldots,9} = [0,1]$
(That big string of unions is supposed to denote breaking up [0,1] into the union of intervals of size 10-N, but I didn't know how exactly to write it... you get the idea though.)

So $t{\notin}I_{0,0,\ldots,0}\wedge t{\notin}I_{0,0,\ldots,0,1} \wedge t{\notin}I_{0,0,\ldots,0,2} \wedge \cdots \wedge t{\notin}I_{9,9,9,\ldots,9}$ implies $t{\notin}[0,1]$, which is a contradiction.

Last edited by a moderator: Mar 22, 2013
2. Mar 22, 2013

Staff: Mentor

You have to be careful with the negation. "For all x exists a sequence such that for all i [...]" negated is "There is a t without a sequence such that for all i [...]", or "There is a t such that for all sequences, there is an i where [...] is wrong."
The problem here: Those intervals for different i are not independent. If you consider some specific i, it is possible to find an appropriate interval, sure. But that restricts the intervals for other i. The proof is still possible, but you have to take care of this. You could consider the first i where t is not in the interval, for example - if there are i with that property, there is a first one as well.