Show dynamical system $([0,1],T_2)$ is exact

Question

Consider the dynamical system $([0,1],T_2)$, where $T_2$ is the tent map given by $$T_2(x)= \begin{cases} 2x & x \in [0,\frac{1}{2}] \\ 2(1-x) & x \in [\frac{1}{2},1] \end{cases}.$$ Let $I \subseteq [0,1]$ be an interval such that $I$ is not a singleton.
I need to show that $([0,1],T_2)$ is exact, i.e. for any non-empty open subset $U \subseteq [0,1]$, there exists some $n \in \mathbb{N}$ such that $T_2^n[U]=[0,1]$.
I have previously shown that if $\frac{1}{2} \notin I$, then there must exist an integer $n \in \mathbb{N}$ such that $\frac{1}{2} \in T_2^n[I]$. I did this by considering the lengths of intervals $T_2^n[U]$ and arriving at a contradiction.
I think I need to use connectedness somehow, but I am not sure how.