According to Wikipedia's article for a regular polygon,
In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed
My question is, how would I actually show this? It makes sense intuitively, but the notion is still somewhat fuzzy and doesn't seem logically airtight. First, I am thinking that I would need some definition of what it means for a sequence of curves or points to converge to some limit.
Taking some inspiration from the standard definition of a limit of a sequence, I came up with the following:
Let $X$ be a metric space. Consider $L \subset X$. (In this specific case, $L$ would be the set of all points in the Euclidean plane that are some distance $r$ from the origin.) Now consider the sequence $x_n$ such that $x_n \subset X,\forall n \in \mathbb{N}$. It can be said that $L$ is the limit of $x_n$ if the following condition holds:
For each real number $\epsilon > 0$, there exists a natural number $N$ such that, for all $n \geq N$, and for all $a \in L$, there exists $b > \in x_n$ such that the distance between $a$ and $b$ is less than $\epsilon$
This seems to work well enough for the specific case of inscribing regular polygons in a circle, as the regular polygons do become arbitrarily close to the circle, but I have some reason to believe it is flawed.
For instance, consider the following sequence: the first term is 1 point placed on a circle. The second term is 2 points placed opposite of each other on a circle. The nth term is n points evenly distributed around a circle. Should the limit of this sequence be exactly equal to a circle?
According to my definition, that seems to be the case. For example, for a circle of radius 1, the furthest a point can be from any members of $x_n$ is strictly less than $\frac{\pi}{n}$. Thus, for any $\epsilon > 0$, choosing $N$ such that $N > \frac{\pi}{\epsilon}$ seems to satisfy my definition unless I'm misunderstanding something.
At first glance, I thought this made sense, but then I realized that it would make more sense if the limit only had a countably infinite number of points, as that is what the sequence seems to approach intuitively. Yet a circle has an uncountably infinite number of points.
I am not quite sure how to reconcile this. Is my definition actually correct here and my understanding is flawed? Does my definition need some amendment to fix it? I imagine it also has some other problems that I am overlooking. I tried searching the internet for a formal definition of this kind of limit, but my google-fu was not strong enough to turn up anything particularly helpful. If anyone has any ideas on how to make my definition more useful, or is able to point me in the right direction, that would be greatly appreciated.