Clarification on Cantor Diagonalization argument? My book is Discrete Mathematics and its Applications.
This is its section on Cantor's Diagonalization argument


I understand the beginning of the method. The author is using a proof by contradiction, saying that assuming a subset of real numbers [0,1] is countable will lead to a contradiction(something that always evaluates to false). He starts listing the real numbers in a countable fashion, r1, r2 and defines dij, come component of the real number) as being some integer in the set of integers from 0 to 9.
Now is where it starts to get confusing for me. I get what this formula is doing 
form a new real number whose ith value will be equal to 4 if rii is not equal to 4 and and 5 if rii is equal to 4. I don't understand the purpose of this real number. He later says that because every real number has a unique decimal expansion(which I totally agree with), r is not equal to r1, r2 because the decimal expansion of r differs from the decimal expansion of ri in the ith place to the right of decimal point, for each i. 
I didnt understand that last part at all. After you form the real number r, say 0.4544 what does it mean for r to have a different expansion of ri, to the ith place to the right of decimal point, for each i. ??
Can someone clarify all of this, make it easier to understand why [0,1]  is not countable?
 A: The new number $r$ has been made in such a way that it is different from each $r_i$ in the $i$th decimal. Let's look at the following example. We have assumed that a list of real numbers between 0 and 1 exists. Such a list may look like the following.


*

*$r_1 = 0.\color{red}123456\dots$

*$r_2 = 0.1\color{brown}35256\dots$

*$r_3 = 0.67\color{green}4523\dots$

*$r_4 = 0.164\color{purple}457\dots$

*$\vdots$


Now, according to the specification of $d_i$, the $i$'th decimal of $r$ must be 4 if the $i$'th decimal of $r_i$ is not 4, and 5 otherwise. So we have that
$$r = 0.\color{red}4\color{brown}4\color{green}5\color{purple}5\dots$$
Notice that $r$ can not be $r_1$ because its 1st decimal is different from the 1st decimal of $r_1$, and it can not be $r_2$ because its 2nd decimal is different from the 2nd decimal of $r_2$ and so on. In general, $r$ can not be the number $r_i$ because it differs from $r_i$ in the $i$'th decimal. Thus, the real number $r$ must be different from every $r_i$, so it is not in the list of all real numbers between 0 and 1 that we assumed existed, and we arrive at a contradiction, and we we therefore have to reject our assumption that it is possible to make a list of the real numbers between 0 and 1 -- in other words, the real numbers are uncountable.
