Union of two Cantor sets is also a Cantor set To prove that a set is Cantor, we have to prove that it's closed and bounded (compact), contains no intervals of positive length, and is perfect. The union of two Cantor sets would also be compact since union of closed bounded sets are still closed and bounded, but I'm having trouble proving whether it contains intervals of positive length or if it's perfect. 
Would the union be a Cantor set? How would I go about proving/disproving it's perfect and contains no intervals of positive length?

Suppose $K_1,K_2$ are Cantor.
That $K_1 \cup K_2$ is perfect: Take a point $x$ in the union. Then $x$ is in one of the sets. Hence it's a limit point of that set. Hence it's a limit point of the union.
No intervals of positive length: Suppose $(a,b) \subset K_1\cup K_2.$ We can choose $x \in (a,b)$ such that $x \not \in K_1.$ Then for $r>0$ small enough, $(x-r,x+r)\cap K_1 = \emptyset$ and $(x-r,x+r) \subset (a,b).$ Hence $(x-r,x+r)\subset K_2,$ contradiction.
