Is it true that $A \in A$? I defined the set $A$ as follow:
\begin{align}
A_0 &amp; =\varnothing \\
A_1 &amp; =\{A_0\}=\{\varnothing\} \\
A_2 &amp; =\{A_1\}=\{\{\varnothing\}\} \\
A_3 &amp; =\{A_2\}=\{\{\{\varnothing\}\}\} \\
&amp; {}\ \ \vdots \\
A &amp; =A_\infty= \{\{\{\cdots\{\varnothing\}\cdots\}\}\}
\end{align}

And now this is clear that $A\in A$.   

Is there any mistake in my conclusion?
I think there are two separate questions here:


Why does the sequence of sets $\{\}, \{\{\}\}, \{\{\{\}\}\}$, . . . not "approach a limit"?
Is it possible to have a set $A$ such that $A\in A$?


The second one is more easily dealt with, since it a precise question. The usual axioms of set theory, ZFC, rule out any such set; specifically, the axiom of Foundation (or Regularity) prevents the existence of such sets. However, if we remove the axiom of Foundation from ZFC, then such sets can exist, and there are even set theories - Quine's NF comes to mind, as well as its better-behaved variant NFU - which require such sets to exist. 

For the former, I think you would find it helpful to try to define in what sense the sequence of sets $\{\}, \{\{\}\}, \{\{\{\}\}\}$, . . . does (in your mind) have a limit, and then try to see what it would take to prove that. Note that there is a clear notion of limit for a monotonic sequence of sets: e.g., if $\alpha&gt;\beta$ implies $A_\alpha\supset A_\beta$ for $\alpha&lt;\kappa$, then it makes sense to define $$\lim_{\alpha\rightarrow\kappa}A_\alpha=\bigcup_{\alpha\in\kappa} A_\alpha,$$ and similarly for descending sequences of sets. But your sequence of sets isn't monotonic, so there's no natural notion of what its "limit" should be. And, even if there were, you would need to argue that properties of terms of the sequence induce properties of the limit, which depends on the properties in question.
Set theory forbids the existence of a set $A$ such that $A \in A$, as that kind of construction easily leads to paradoxes, such as Russell's paradox.

In ZFC, the axiom of regularity has this immediate consequence.