I saw a false proof meme on Reddit that shows $\pi = 4$. I'm adding the meme here. Pardon my ignorance, but I don't quite understand why this proof is wrong. 3Blue1Brown made a video in which he talks about this wrong proof. Here's an explanation of 3Blue1Brown on why the proof is wrong.
He symbolises the unfolded square as $c_0$ and the shape formed after n folds as $c_n$.Based on these symbols, we can find out the following facts: $$\begin{align} \forall n\in \mathbb N, len(c_n)=len(c_{n+1}) \tag{1}\label{eq1} \\ \forall n\in \mathbb N, len(c_n)=len(c_0) \tag{2}\label{eq2} \\ \lim_{n\to\infty}len(c_n) = len(c_0) \tag{3}\label{eq3} \end{align}$$
What 3Blue1Brown says is that the representation of the circle with these symbols will be $\lim_{n\to\infty}c_n$, and its length will be $len(\lim_{n\to\infty}c_n)$. And he finishes the disproof with saying $\lim_{n\to\infty}len(c_n)\neq len(\lim_{n\to\infty}c_n).$
I understand this much, it makes perfect sense. However, I couldn't quite understand why this last inequality is true. Can you prove this inequality? I'm sorry if this newbie question is a bit too low-level for this professional platform, but I'm really asking to fully understand. Thank you.
