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Assume that $a$, $b$ are real numbers. Prove that there exists integers $m$ and $n$ that satisfy $a < \frac{m}{2^n} < b$.

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Let $x=b-a$. Then $2^nb-2^na=2^nx$. Choose $n$ big enough to make $2^nx>1$. Then there is an integer, $m$, such that $2^na<m<2^nb$.

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