Why does one third belong to the Cantor set?

Question

We know that all numbers that belong to the Cantor set have a ternary representation with only 0's and 2's but, for example, $\frac{1}{3}=(0.1)_3$ and $\frac{1}{3}$ belong to the Cantor set. I don't understand and I have already read tons of sources. Can somebody explain to me what is going on?

Answer 1

$(0.1)_3 = (0.0222\ldots)_3$, just as $0.1 = 0.0999\ldots$ in base $10$.

Answer 2

$0.02222222222222222222..._3 = 0.1_3$

Answer 3

Any endpoint of removed intervals belongs to Cantor set.