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The question is: Suppose $\alpha>0$. Prove that any real number $x$ can be written uniquely in the form $x=k\alpha+x'$, where $k$ is an integer and $0\leq x'\lt\alpha$.

How do I approach this problem? Perhaps construct a set for all $x$ that cannot be expressed in this way and show that it is empty?

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Hint: it is sufficient to show that any real number can be written as a sum of its integer and "fractional" or "decimal" parts.

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