Comparing Infinite Sets Hello Stack Exchange Community, 
I just read a book on Cantor and how he proved that the real numbers were a larger infinity than the natural numbers. He uses bijection and claims that if you were to write out all of the natural numbers you would be able to biject every single one with a real number, but after this bijection there would still be more real numbers. However, could you also use the same logic and biject all the real numbers with natural numbers and when you have a new real number, just add 1 to your previous real number? Also, since we can list all the fractions by listing 1 over every natural number, then 2 and 3 and on to infinity, couldn't we just write the "first" real number by just writing inifitely many zeroes, then a 1, then repeat but write 2, then 3, and all the natural numbers? 
This may just be a conceptual thing but I don't understand Cantor's logic and the different types of infinities.
 A: just because you never stop doesn't mean your process is taking you "closer" to the solution.
Your proposed method is similar to a painter who wants to paint an infinite plane, and all he does is paint a straight red line, which he keeps painting for all eternity. Yes, he is always painting new parts, but anyone can tell that this method won't work if you want to make the whole plane  red.
Exactly the same is happening here, you're always adding "paint" (by pairing up one extra real at a time) but you're never even gonna get close to finishing.
A: What Cantor's proof shows is that there is no surjection (onto mapping) of the natural numbers onto the real numbers. The proof considers a function, $f : \Bbb{N}\to\Bbb{R}$ and exhibits a real number $x$ that cannot be in the range of $f$, by choosing the $n$-th digit in the decimal expansion of $x$ to be different from the $n$-th digit in the decimal expansion of $f(x)$. This argument relies on the fact that the domain of $f$ comprises the natural numbers: it uses the natural number $n$ to identify something about $f(n)$ (namely its $n$-th decimal digit). The technique doesn't generalise to functions on the real numbers.
