I've read that the set of real numbers *R* is uncountable. It was proved by contradiction. A number *x* that is not in the *f(n)* side was constructed. 
Ultimately it was said that "*x* is not *f(n)* for any *n* because it differs from *f(n)* in the *n*th fractional digit.
I can't understand the last part. I understand what is being said but don't see how it proves that *R* is uncountable.



Uncountable Sets
Assume the interval $[0,1)$ is countable. Then we can write down all numbers like that:


* $a\_0 = 0.a\_{0,0}a\_{0,1}a\_{0,2}\dots$
* $a\_1 = 0.a\_{1,0}a\_{1,1}a\_{1,2}\dots$
* $a\_2 = 0.a\_{2,0}a\_{2,1}a\_{2,2}\dots$
* $\dots$


Now we construct a new number $b=0.b\_0b\_1b\_2\dots$ in the following way. We look at $a\_{ii}$. If this digit is $1$, then $b\_i=2$, otherwise we set $b\_i=1$. Therefore we constructed a now number, that is not in our list of $a\_i$, since it differs from each number in one position. But the number $b$ is clearly in the interval $[0,1)$ and we assumed, that we have written down all numbers. Contradiction, i.e., the set was not countable.


And since $[0,1) \subseteq \mathbb{R}$, the real numbers are also not countable.


**Edit.** Using the notation of the original posting: $b=x$ and $f(n)=a\_n$.


