A proof that the Cantor set is Perfect

Your idea is sound, but you’ve not expressed it clearly enough for you to have a real proof. I’ll write up an argument along the general lines that you have in mind.

Let $x\in\Delta$ and $\epsilon>0$ be arbitrary. Choose $n\in\Bbb N$ large enough so that $3^{-n}<\epsilon$. $C_n$, the $n$-th stage in the standard construction of $\Delta$, is the union of $2^n$ pairwise disjoint closed intervals, each of length $3^{-n}$; let $I$ be the one of these intervals containing $x$, clearly $I\subseteq B_\epsilon(x)$.

Now consider $C_{n+1}$: it’s a disjoint union of $2^{n+1}$ closed intervals, each of length $3^{-(n+1)}$, and exactly two of these intervals, say $I_0$ and $I_1$, are subsets of $I$. Let $I_0$ be the one that contains $x$. $\Delta\cap I_1$ is non-empty for the same reason that $\Delta$ is non-empty (why is that?), so let $y\in\Delta\cap I_1$. Then $y\in\Delta\cap\big(B_\epsilon(x)\setminus\{x\}\big)$, and since $\epsilon>0$ was arbitrary, $x$ is a limit point of $\Delta$. Finally, $x\in\Delta$ was arbitrary, and $\Delta$ is closed, so $\Delta$ is perfect. $\dashv$

It isn’t actually necessary to split $I$ into $I_0$ and $I_1$. Let $I=[a,b]$; then $a,b\in\Delta\cap B_\epsilon(x)$, and since $x$ cannot be equal to both $a$ and $b$, $\Delta\cap\big(B_\epsilon(x)\setminus\{x\}\big)\ne\varnothing$.

Let $x\in \Delta$, For any $\epsilon>0$, consider $(x-\epsilon,x+\epsilon)$ Using archemedean propery, $\exists N\in \mathbb N: \frac{1}{3^N}<\epsilon$. Let $\frac{M}{3^N}=\max \{\frac{m}{3^N}:\frac{m}{3^N}<x \space \text{and} \space m\in \mathbb N\}$. So $x\in [\frac{M}{3^N},\frac{M+1}{3^N}]\subset (x-\epsilon,x+\epsilon).$ Consider the $N+1$ the stage of construction Removing $(\frac{3M+1}{3^{N+1}},\frac{3M+2}{3^{N+1}})$ from $[\frac{M}{3^N},\frac{M+1}{3^N}]$.

Case 1 $x\in [\frac{M}{3^N},\frac{3M+1}{3^{N+1}}].$ there exists $c\in \Delta$: $c\in [\frac{3M+2}{3^{N+1}},\frac{M+1}{3^{N}}]$. Hence $(x-\epsilon,x+\epsilon)\cap \Delta \setminus \{x\}\neq \emptyset.$ So, $x$ is a limit point of $\Delta$.

Similarly

Case 2 suppose $x\in [\frac{3M+2}{3^{N+1}},\frac{M+1}{3^{N}}]$, We have $(x-\epsilon,x+\epsilon)\cap \Delta \setminus \{x\}\neq \emptyset.$ Hence, $x$ is the limit point of $\Delta$. So, $\Delta$ is a perfect set.