Let's say we have $\mathbb{N}$, the set of natural numbers:
$\{1, 2, 3, 4, 5...\}$
...which has a cardinality of infinity, and the set $A_x$ which consists of the variable "$x$" (so $\{x\}$).
If I did this:
$N - A_1 - A_2 - A_3 - \cdots - A_x$
...for the limit as $x \rightarrow \infty$, then does the resulting set have a cardinality of $0$? Or does the fact that $x$ is a limit mean that it is not the same type of infinity and that the remaining set will still have a cardinality of infinity?
On a similar note:
Is the cardinality of the set of natural numbers the same as the cardinality of the set: $\{x: x$ is an integer such that $\displaystyle 0 < x < \lim_{n \rightarrow \infty} n\}$ ?