Limit of the sequence of regular n-gons. Let $A_n$ be the regular $n$-gon inscribed in the unit circle.
It appears intuitively obvious that as $n$ grows, the resulting polygon approximates a circle ever closer. 
Can it be shown that the limit as $n \rightarrow \infty $ of $A_n$ is a circle?

Given a sequence of sets $(A_n)_{n\geq3}$ there is a natural $\lim\inf_{n\to\infty} A_n=:\underline{A}$ and a natural $\lim\sup_{n\to\infty}A_n=:\overline{A}$ of this sequence. 
In the problem at hand the $A_n$ are closed regular $n$-gons inscribed in the unit circle, all sharing the point $P:=(1,0)$.
The set $\underline{A}$ consists of all points that are in all but finitely many of the $A_n$. It is easy to see that  all points $z\in D:=\{(x,y)\ |\ x^2+y^2 < 1\}$ satisfy this condition and that in fact $\underline{A}=D\cup\{P\}$.
The set $\overline{A}$ consists of all points that are in infinitely many $A_n$. Obviously $\overline{A}\supset\underline{A}\ $, and $\overline{A}$ is contained in $\overline{D}=\{(x,y)\ |\ x^2+y^2 \leq 1\}$. In fact $\overline{A}\cap\partial D$ consists of all points on the unit circle whose argument is a rational multiple of $\pi$.
This is how much you can say on the pure set-theoretical level; an actual limit set $A_*$ does not exist. 
