Each point in Cantor Set can be built according to the well-known "delete the middle third" rule but also as a real number in the unit interval:
In arithmetical terms, the Cantor set consists of all real numbers of
the unit interval $[0,1]$ that do not
require the digit $1$ in order to be expressed as a ternary (base 3)
fraction.
(Source: Cantor Set on Wikipedia)
It is also known that when writing numbers from an infinite sequence of digits in a given base, infinitely repeating the highest digit leads to an alternate way of writing a number having a finite number of digits in the same base. Thus, I expect $0.0222\dots_3=0.1_3$.
Now, I am unsure whether $0.0222\dots_3$ actually belongs to Cantor set for obvious reasons: thinking at the number as $0.0222\dots_3$ makes me think it belongs to the set, as it describes a perfectly valid path on the picture below (path being: left, then always right); on the other hand, the number $0.1$ obviously does not belong to the set. Then, does $0.0222\dots_3$ belong to the set?
Best Answer
Yes, the number $0.022222..._3 = 0.1_3 = \frac{1}{3}$ lies in the Cantor set. In each step of the Cantor set you take out the open interval in the middle, so you get
Step $0$: $[0,1]$
Step $1$: $[0,\frac{1}{3}]\cup[\frac{2}{3},1]$
...
and after Step 1, you never change anything near $\frac{1}{3}$, so it remains inside for all steps and thus is in there at the end as well.
This is also consistent with the Wikipedia comment, the important detail is: that do not require the digit 1.
$0.1_3$ does not require the digit $1$, since $0.022222..._3 = 0.1_3$ and thus (according to this rule) is part of the cantor set.