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Questions tagged [floor-function]

The floor function, also known as the greatest integer function, maps a real number $x$ to the greatest integer less than or equal to $x$ (often denoted $\lfloor x \rfloor$). See also (ceiling-function).

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Prove $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{n}+\sqrt{n+2}]$ for all integers $n\ge 1$

This is the full question Prove that of the two equations $$[\sqrt{n}+\sqrt{n+1}] = [\sqrt{n}+\sqrt{n+2}] \\ [\sqrt{n}+\sqrt{n+1}] = [\sqrt{n}+\sqrt{n+2}]$$ the first one holds for ...
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Prove that $\sum_{j=1}^{\infty} \left[ \frac{n}{2^j}+\frac{1}{2} \right] = n$ for interger $n\ge 1$

This is the full question: Consider an integer $n\ge 1$ and the integers $i$,$1\le i \le n$. For each $k=0,1,2,\cdots$, find the number of $i$'s that are divisble by $2^k$ but not by $2^{k+1}$.Thus ...
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Proving Infinite Limit from Definition, Floor Function

I want to formally prove that the function $$f(x)=\textrm{floor}\left(\frac{1}{\pi}\left(x-\frac{\pi}{2}\right)\right)$$ tends to infinity as $x\rightarrow\infty$. More specifically, I am having ...
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Floor function of a non-integer Cauchy sequence also a Cauchy sequence?

Let $a_n$ be a Cauchy sequence such that $a_n$ converges to a non-integer value. Then if we have a sequence $b_n$ defined as $b_n \leq a_n < b_n+1$, will $b_n$ be a Cauchy sequence? Intuitively, I ...
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Let x ∈ R. Show that $2\lfloor x \rfloor \leq \lfloor 2x \rfloor \leq 2\lfloor x \rfloor + 1$

I already checked some questions about this statement, however I can't understand why the first inequality is true. We know that $\lfloor x \rfloor \leq x$, then $2\lfloor x \rfloor \leq 2x$. In ...
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Limit of a greatest integer function (sided limit) [closed]

What is the value of $\lim\limits_{x\to 0^+} \dfrac{b}{x}\left\lfloor\dfrac{x}{a}\right\rfloor$ for $a>0$ and $b>0$. Note that $\lfloor x\rfloor$ denotes the greatest integer less than or equal ...
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Prove $⌊2x⌋+⌊2y⌋≥⌊x⌋+⌊y⌋+⌊x+y⌋$

Prove the following for all real $x$ i. $⌊2x⌋+⌊2y⌋≥⌊x⌋+⌊y⌋+⌊x+y⌋$ ii. $⌊x⌋-2⌊x/2⌋$ is equal to either $0$ or $1$ For ($ii$.) I attempted to split it into cases of whether the fraction part {$x$} is ...