# Can somebody explain why the interval $\left ( 0,1 \right )$ is not countable? [duplicate]

I cannot seem to understand the proof of why the interval $\left ( 0,1 \right )$ is not countable.

The proof that is written in my book using the method of Reductio ad absurdum.

It starts with the following statement:

We know that every real number can be written as a decimal.

Let $x_{1} = 0,x_{11}x_{12}x_{13}...$

$x_{2} = 0,x_{21}x_{22}x_{23}...$

$x_{3} = 0,x_{31}x_{32}x_{33}...$

Then by using the diagonial argument it constructs a $y = 0,y_{i}...$ but cannot understand the process that follows.

## marked as duplicate by Asaf Karagila♦, Hanul Jeon, Lord_Farin, MathOverview, Dan RustNov 11 '13 at 12:22

• The point here is that the supposed list $x_1$, $x_2$, ... is <i>assumes</i> to be countable. The diagonal argument constructs $y$ which is different from every $x_i$, and thus not in the list. So the interval cannot be countable. – Jas Ter Nov 11 '13 at 11:11
• Can you elaborate a little more o nthe diagonal argument please? – Rrjrjtlokrthjji Nov 11 '13 at 11:21
• If it's countable, its Lebesgue measure must be $0$. This is absurd because its measure is $1$. – Makoto Kato Nov 11 '13 at 11:27
• Sorry but I have no idea what a Lebesgue measure is... I am a first year student. – Rrjrjtlokrthjji Nov 11 '13 at 11:31
• @Nick : This argument is called "Cantor's diagonalization". Google that and you should find plenty of information. – Prahlad Vaidyanathan Nov 11 '13 at 11:32

For each $x_{ii}$, you pick $y_i$ such that $x_{ii}\neq y_i$, and of course, $y_i$ can only take values from $0,1,2,3,4,5,6,7,8,9$.
Then you set $y=0,y_1y_2y_3\dots$, and you'll note that $x_i\neq y$ for all $x_i$. So there is no bijection between $(0,1)$ and $\mathbb N$.
But of course this argument doesn't quite work, because what happens if $x_1=0.500\dots$, and $y_1=4$, $y_i=9$ for all $i\geq 2$. Then I would get $y=0.4999\dots=0.5=x_1$. So to fix this, simply don't let $y_i$ take the values $0$ or $9$. That's because a number has two decimal representations if and only if its decimal representation has repeating $9$ or repeating $0$ after some point.