A proof that the Cantor set is Perfect

Question

I found in a book a proof that the Cantor Set $\Delta$ is perfect, however I would like to know if "my proof" does the job in the same way.

Theorem: The Cantor Set $\Delta$ is perfect.

Proof: Let $x \in \Delta$ and fix $\epsilon > 0$. Then, we can take a $n_0 = n$ sufficiently large to have $\epsilon > 1/3^{n_0}$. Thus, the interval $[a, b]$ where $x$ lies is a subset of $B_\epsilon > (x)$. Hence, by iterating the construction of the Cantor set for $N > n_0$, we have intervals of length $1/3^N$ all included in $B_\epsilon (x)$, but with only one of those intervals such that $x$ lies within.

The intution behind the proof was that we should prove that for every $x$, if $x \in \Delta$, then for every $\epsilon >0$, $B_\epsilon (x) \setminus \{x\} \cap \Delta \neq \varnothing$.

Now, I do not particularly like my reference to the $[a, b]$ interval that is not mentioned before. Moreover, here – by choosing a closed interval – I am trying to address all at once the case in which $x$ is an endpoint of one of the closed intervals that form $\Delta$. Finally, I did not close the proof with a statement like "Thus, there are infinitely many points that differ from $x$ and that lie within $B_\epsilon (x)$.

In the end, I am not completely sure if this can be considered a proof or not. The intuition is correct (I am kind of positive about it), but I am not sure if I was actually able to write down my intuition in a good way.

As always any feedback is more than welcome.
Thank you!

Answer 1

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 $b$, $\Delta\cap\big(B_\epsilon(x)\setminus\{x\}\big)\ne\varnothing$.