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I have a question in my text that asks me to prove that if $x- \lfloor x\rfloor \geq \frac{1}{2}$, then the $\lfloor 2x\rfloor=2\lfloor x\rfloor+1$.

I understand the proof up to the point where they obtain that:

2*floor of x+1 is less then or equal to 2x is less then or equal to 2*the floor of x+2. But then it simply states, by the definition of floor, the desired conclusion follows. Can anyone explain why this is in some detail? I know what floor means, but perhaps i'm just not seeing something small

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    Note that the phrase "2 times floor of x + 1" is ambiguous, meaning either $2\lfloor x+1\rfloor$ or $2\lfloor x\rfloor +1$. The problem means the latter.2012-07-25

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