The Cantor set C can also be defined as follows: Let $ A_{0}:=[0,1] $ and $ A_{k}:=\frac{1}{3} A_{k-1} \cup \frac{1}{3}\left(A_{k-1}+2\right) $ for $ k \in \mathbb{N} $. Then $ C=\bigcap_{k=0}^{\infty} A_{k} $.
Now I am to show that the union occurring in the definition of $ A_{k} $ is disjoint, and that $ A_{k} \subset A_{k-1} $ holds.
Disjunct means that two sets do not have a common element. Thus I would have to show that $ A_{k}:=\frac{1}{3} A_{k-1}$ and $\frac{1}{3}\left(A_{k-1}+2\right) $ have no common element?
Unfortunately, it is not clear to me how I can show disjointness in this case ( maybe a proof of contradiction by assuming that one element occurs in both would be a possibility? - how would this then be shown?)