How do I know that this R number is different to all numbers in your sequence? Well, I can take any M number from your sequence and compare it to R. However, the Mth number is different to R in its Mth digit.
About the limit in the infinite... it doesn't matter at all... You can't take a M value that's infinite, as we have only numbers that have a finite size (there are infinite numbers with finite size, that's the trick on this; there aren't integers with an infinite size for example). And if you take an M that's finite, then you have to say that such Mth element is different to my R.
And so on for any sequence you can build (even for sequences you can't build, like random sequences).
Edit: possibly the big thing that's confusing you is that you can't have an infinite number, even if there are infinite numbers. So, taking the limit of a function is something that has a very restrict meaning that is actually not directly related to the concept of infinity. If you check the definition of limit it doesn't talk about infinities but about deltas, and if you use such definition your proposed problem stops being a problem. http://en.wikipedia.org/wiki/Limit_(mathematics)#Formal_definition