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!