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For $n\geq 2$, let $m_n:\mathbb{R}/\mathbb{Z}\rightarrow \mathbb{R}/\mathbb{Z}$ be the map $m_n(x)=nx (\mod \mathbb{Z})$. Cover the circle $\mathbb{R}/\mathbb{Z}$ by $n$ non overlaping closed intervals $I_\alpha=[\alpha/n,(\alpha+1)/n]$ for $\alpha\in \{0,1,...,n-1\}$. Given any sequence $\alpha_0,\alpha_1,...$ with $\alpha_i\in \{0,1,...,n-1\}$ show that there is a number $x_0=0.\alpha_0\alpha_1...(base \: n)=\sum \alpha_i/n^{i+1}$ in $\mathbb{R}/\mathbb{Z}$ with the property that the orbit $m_n:x_0\rightarrow x_1 \rightarrow x_2 \rightarrow ...$ of $x_0$ under $m_n$ satisfies $x_k\in I_{\alpha_k}$ for every $k\geq 0$.

I am new to Dynamical systems. Also I am not quite sure what I have to show in this problem. Given any such sequence they have already defined $x_0$ so what do we need to prove? Do we need to prove that, that number satisfy the latter property? Or else what?

I have no clue what to do here. Could somebody please help me to proceed?

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    yes, prove that the number they define does satisfy the property $m_n^k(x_0) \in I_{\alpha_k}$ (there's not much to do). A more natural but more difficult question would have been not to give you $x_0$ and to make you find it – 2017-01-18
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    The intervals are overlapping. – 2017-01-23

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