Homework - set theory infinite union A question from my homework I'm having trouble understanding.
We are given:
$A(1) = \{\varnothing\}$, $A(n+1) = A(n)\cup (A(n)\times A(n))$
$A=A(1)\cup A(2)\cup A(3)\cup \cdots \cup A(n)\cup A(n+1) \cup \cdots$ to infinity
The questions are:
1) show that $A\times A \subseteq A$
2) Is $A \times A = A$?
Thank you for your help.
I've tried writing $A(2)$ but it gets really complicated and I'm having trouble understanding what the sets are. Let alone solve the question.

HINT: For (2), note that not all the elements of $A$ are ordered pairs.

Also, let's write $A(2)$, but to make it simpler let's call $A(1)=X$. Then $A(2)=X\cup(X\times X)=\{\varnothing\}\cup\{\langle\varnothing,\varnothing\rangle\}=\{\varnothing,\langle\varnothing,\varnothing\rangle\}$.
Not very difficult, $A(3)$ will have six elements.
