TITLE: A proof that the Cantor set is Perfect
QUESTION [13 upvotes]: 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!

REPLY [5 votes]: 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.