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#### Alexmahone

##### Active member

- Jan 26, 2012

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Let $\alpha$ be a real number between 0 and 1 written in binary: e.g.,

$\alpha=.1011001\ldots$ means $\alpha=\frac{1}{2}+\frac{1}{2^3}+\frac{1}{2^4}$$+\frac{1}{2^7}+\ldots$

Make a set of nested intervals by starting with $I_0=[0,\ 1]$, then defining recursively $I_n$ to be the (closed) left half of $I_{n-1}$ if the $n$-th place of $\alpha$ is 0, and the (closed) right half if the $n$-th place is 1.

Prove that the resulting sequence of nested intervals converges to $\alpha$, i.e., $\alpha$ is the unique number inside all the intervals.

$\alpha=.1011001\ldots$ means $\alpha=\frac{1}{2}+\frac{1}{2^3}+\frac{1}{2^4}$$+\frac{1}{2^7}+\ldots$

Make a set of nested intervals by starting with $I_0=[0,\ 1]$, then defining recursively $I_n$ to be the (closed) left half of $I_{n-1}$ if the $n$-th place of $\alpha$ is 0, and the (closed) right half if the $n$-th place is 1.

Prove that the resulting sequence of nested intervals converges to $\alpha$, i.e., $\alpha$ is the unique number inside all the intervals.

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