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May I know the standard proof technique to prove such kind of inequalities.

$2 \lfloor x \rfloor \leq \lfloor 2x \rfloor \leq 2 \lfloor x \rfloor +1$

Thanks!

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    See also: http://math.stackexchange.com/questions/976665/show-that-0-leq-left-lfloor-frac2ab-right-rfloor-2-left-lfloor2016-08-28

2 Answers 2

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By Hermite's identity, we know that $ \lfloor x \rfloor + \lfloor x \rfloor \le \lfloor x \rfloor + \lfloor x + \frac 12 \rfloor = \lfloor 2x \rfloor \le \lfloor x \rfloor + \lfloor x + 1 \rfloor$. Alternatively, as already mentioned, you can use casework on $\{x\} := x - \lfloor x \rfloor$, in particular when $0 \le \{x\} < 1/2$ and when $1/2 \le \{x\} < 1$.

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    I'd imagine you figured it out by now, but suppose $x = n + r$ where $n \in \mathbb{Z}$ and 1/2 \le r < 1. Then $2\lfloor x \rfloor = 2n, \lfloor 2x \rfloor = 2n+1, 2\lfloor x \rfloor + 1 = 2n+1$.2011-08-30
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Hint: let $n = \lfloor{x\rfloor}$, so $n \le x < n+1$. What about $2x$?