cantor set prove verfication

Question

Question:

My proof:

Can anyone help me verify my proof?

Answer 1

This proof is correct (apart from, as noted in comments, the first line on the $2$nd page should still say $x\in C$ as opposed to $x\in C^c$).

I would probably simplify the first part of the proof by always choosing the "left" end of the subinterval, rather than caring a lot whether to take the left or the right end.

Commentary: In essence, if you interpret Cantor set as the real numbers in $[0,1]$ written in the ternary system without using the digit $1$, then, for $x\in C$ you have the ternary expansion such as:

$$x=0.a_1a_2\ldots a_{n-1}a_na_{n+1}\ldots$$

where $a_i\in\{0,2\}$, and in the first case you can use:

$$x_n=0.a_1a_2\ldots a_{n-1}a_n$$

(a finite expansion - that is what I am suggesting above), or:

$$x_n=0.a_1a_2\ldots a_{n-1}a_na_na_n\ldots$$

(which is what you are doing in your original proof). In the second case, you are using

$$x_n=0.a_1a_2\ldots a_{n-1}111\ldots$$

which is perfectly ok.