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

## Homework Statement

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

## Homework 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 $I_{k_1,k_2,\ldots,k_n}$

## 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.