Question about an example of topologocally mixing map

Question

How to show that the doubling map $T:[0,1) \to [0,1)$ given by $T =2x $ (mod $1$) is topologically mixing?

Topologically mixing means that for any pair of open sets $U,V$ there is a large $N$ such that $T^n(U)\cap V\neq\emptyset$ for all $n\geq N$

Answer 1

For each $x\in [0,1)$ and each positive integer $n$ we have $T^n(x)=2^nx (\operatorname{mod} 1)$. There are non-negative integers $N$ and $i<2^N$ such that $W=[i2^{-N},(i+1)2^{-N})\subset U$. Then $T^n(U)\supset T^n(W)=[0,1)$.