Cantor’s diagonal proof revisited In his diagonal argument (although I believe he originally presented another proof to the same end) Cantor allows himself to manipulate the number he is checking for (as opposed to check for a fixed number such as $\pi$), and I wonder if that involves some meta-mathematical issues. 
Let me similarly check whether a number I define is among the natural numbers. The number is $n+1$ and it is clear that $1$, $2$, $3,\ldots,\ n$ are not among these numbers. This “proves” that $n+1$ is not a natural number.
I have here, just like Cantor, a formula for a number, rather than a given number. What is the difference between our proofs?
It seems to me that the answer is that Cantor’s number (as opposed to mine) is being successively better bounded; the process of moving forward in his enumeration describes a converging series, such as we use to define the real numbers, using Cauchy limits. You don’t hear this added comment in the proof. Don’t you think it belongs to the proof? 

Cantor's proof is way more general than just showing that $\mathbb R$ is uncountable. The diagonal argument actually shows that no set can have the same cardinality as its power set (i.e. the set of all its subsets). 
Regarding the concrete case of the real numbers, already when you state that real numbers can be written using infinite decimal expansions, you are using that they are characterized via a convergent series. No need to keep repeating that. 
As for your argument, I cannot see what you mean when you say "not among those numbers". 
