Given two sets $A$ and $B$, I want to know the result of: $C = A \setminus B$
If I define the sets $A(n)$ as the set such that: $\begin{align*} A(n) \setminus B &= \emptyset; &\quad &\forall n= 0,1,2,3,\ldots,\infty &\quad &(1)\\\ \lim_{n\to\infty} A(n) &= A &&&&(2) \end{align*}$
If using $\lim\limits_{n\to\infty}$ over a set $A(n)$ is not possible, how can I describe the process of iterating over $n$ up to $\infty$?
Read then $\lim\limits_{n\to\infty}$ as "when $n$ approaches to $\infty$".
Then:
$ \lim_{n\to\infty} \quad A(n) \setminus B \quad = \quad \emptyset, \quad \quad \quad \text{by} \quad (1)$
but also,
$\lim_{n\to\infty} \quad A(n) \setminus B \quad = \quad A \setminus B,\quad \text{by} \quad (2)$
So, is then:
$ A \setminus B = \emptyset{}\quad \quad?$ That is: $ C=\emptyset \quad? $