Inverse limit

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Suppose that $I_n=[a_n,b_n]$ is a sequence of intervals and for each positive integer $n$ there is a function $f_n \colon I_{n+1} \rightarrow I_n$. Let $\vec{f}=(f_1,f_2,\ldots)$. We define the inverse limit $\varprojlim \vec{f}$ to be the set of all sequences $\vec{x}=(x_1,x_2,\ldots)$ with the property that $x_i = f_i(x_{i+1})$ for each positive integer $i$.