I'm working on a problem from the book "Introduction to Topology" by Bert Mendelson:
If $A_1\subset A_2, A_2\subset A_3, \ldots , A_{n-1}\subset A_n$, and $A_n \subset A_1$, prove that $A_1=A_2=\cdots=A_n$.
I know how to prove this, but my question is how rigorous my proof should be. For example, to make my proof easier, I proved the following "lemma":
If $H$ and $J$ are sets, $H \subset J$, and $J\subset H$, then $H=J$.
My proof went like this:
From the givens, we can determine that $$\alpha \in J, \forall \alpha \in H$$ $$\beta \in H, \forall \beta \in J$$ Which means that $$\alpha \in J, \forall \alpha \in H$$ $$\neg \beta \notin H, \forall \beta \in J$$ and so $H=J$.
I then went on to prove that $$A_k\subset A_{k+1}, A_{k+1}\subset A_k, \forall k \le n$$
Is my "lemma" proof enough of a proof? This is such a basic lemma that it seems like it should be obvious... but then again, when something seems obvious, it sometimes isn't. Is this rigorous enough? Is it too rigorous?