# Infinite sequences and uncountability

• Dec 11th 2009, 01:31 PM
dude15129
Infinite sequences and uncountability
So, I am having trouble with this proof and am not sure I understand.

Use Cantor's diagonal argument to show that the set of all infinite sequences of 0s and 1s (that is, of all expressions such as 11010001...) is uncountable.

I definitely understand what Cantor's diagonal argument is and I know what uncountable is, but I guess I just don't understand what or how to do the infinite sequences of 0s and 1s.

Thanks!
• Dec 11th 2009, 01:45 PM
artvandalay11
how did you see the diagonal argument with "yes" and "no"? basically you had functions spitting out 2 values and you just make them 0s and 1s. Since every real number has a binary representation (i.e. a representation as 0s and 1s) the set of all infinite sequences of them (real numbers) is uncountable
• Dec 11th 2009, 01:51 PM
dude15129
I'm not sure what you're asking or saying.

I just don't understand this infinite sequence of 0s and 1s. I know that every number can be written as 0s and 1s but i don't understand it.

Sorry.
• Dec 11th 2009, 01:59 PM
Plato
Quote:

Originally Posted by dude15129
I definitely understand what Cantor's diagonal argument is and I know what uncountable is, but I guess I just don't understand what or how to do the infinite sequences of 0s and 1s.

Suppose that the set is countable.
So each sequence in the set can be named with a unique positive integer.
Then $x_n$ is made upthis way: $x_{n,m}$ is the $m^{th}$ term in the sequence named $n$.
Go through the change each diagonal element to the oppsite.
• Dec 11th 2009, 02:18 PM
artvandalay11
you said you understood the diagonal argument, so you were looking at a table, what values did you have in each table cell, but just use Plato's post, surely he is better than I