We're going to take a cue from our first example and consider
infinite strings of 0's and 1's. In this case, we make
use of the fact that each element of [0,1] can be written as a
binary decimal. That is, a number of the form
Example 2
The interval [0,1] (in R) is an uncountable set.
We're going to take a cue from our first example and consider
infinite strings of 0's and 1's. In this case, we make
use of the fact that each element of [0,1] can be written as a
binary decimal. That is, a number of the form
More generally, each 
in [0,1] can be written as
where each 
is either 0 or 1.
Here are a couple of easy examples:
(How would you check the second example?)
Now just like ordinary decimals, a given number may have
more than one binary decimal representation. For instance,
But this minor inconvenience won't effect our arguments here. In order to be precise, let's agree to always take the representation ending in an (infinite) string of repeating 1's whenever there's a choice.
The point here is that the elements of [0,1] can be thought of as
infinite length strings (or sequences) of 0's and 1's. The claim
here is that the collection of all infinite sequences of 0's and 1's
cannot be counted.