- #1

- 71

- 0

## Homework Statement

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

## Homework Equations

Let [itex]x\in{[0,1]}[/itex], then the decimal expansion for x is an infinite sequence [itex](k_{i})^{\infty}_{i=1}[/itex] such that for all i, [itex]k_i[/itex] is an integer between 0 and 9 and such that [itex]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][/itex].

We call that interval above [itex]I_{k_1,k_2,\ldots,k_n}[/itex]

## The Attempt at a Solution

Assume BWOC that there exists a real number [itex]t\in[0,1][/itex] with no decimal expansion. That means there exists an [itex]N\in{\mathbb{Z}}[/itex] such that for all sequences [itex](k_{i})^{N}_{i=1}[/itex], [itex]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][/itex].

But [itex]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][/itex]

(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 [itex]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}[/itex] implies [itex]t{\notin}[0,1][/itex], which is a contradiction.

How does this look? Advice?

Last edited by a moderator: