# Tagged Questions

The function that maps a real number $x$ to the largest integer less than or equal to $x$ (often denoted by $\lfloor x\rfloor$). See also (ceiling-function).

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### Floor and ceiling opposite property

For $x\in \mathbb{R}$ let's define $[x]$ as: $$[x] = max \{ k\in \mathbb{Z}: k\leq x \}$$ and $[x]^{*}$ as: $$[x]^{*} = min \{ k\in \mathbb{Z}: k\geq x \}.$$ Show that: $$[x]^{*} = -[-x].$$ So ...
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### How does a C-constant in a substitution affect the result of integration?

I'm doing some pretty strange integrals (floor functions ones) and I think I should probably start asking some more complex questions regarding it. Since I now know how to integrate them, I have to ...
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### Need to verify Real Analysis Proof

I wish to verify a proof; solutions to this exercise are not available. Lemma: If $S \subset \mathbb{Z}$ is bounded from above, it has a maximum element. Proof: If $S$ is bounded from above, it ...
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### Summation of $\sum_{k=1}^{n}\left \lfloor \log _{m}k \right \rfloor$ and $\sum_{k=1}^{n}\left \lceil log_{m}k\right \rceil$

$$\sum_{k=1}^{n}\left \lfloor \log _{m}k \right \rfloor$$ $$\sum_{k=1}^{n}\left \lceil log_{m}k\right \rceil$$ I found myself stuck trying to solve these two summations but i can't make any progress. ...
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### Limit of the floor function of $\frac{x}{\sin(x)}$

Alright this looks like a very simple problem at the first go. I need to find $$\lim_{x\rightarrow0^+} { \left\lfloor{\frac{x}{\sin (x)}}\right\rfloor}$$ So since I know the inner function's graph ...
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### Proving associative property, floor function

I need to prove the following operation is associative: $x*y = xy \pmod 5$ I came up with the equation that $x*y=xy-5[\![xy/5]\!]$ I'm having difficulty proving that $x*(yz)=(xy)*z$. After ...
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### $\lim_{x\to \infty}\frac{3}{x}\lfloor\frac{x}{4}\rfloor=\frac{3}{4}$

Prove that $\lim_{x\to \infty}\frac{3}{x}\lfloor\frac{x}{4}\rfloor=\frac{3}{4}$ If i put $x\to\infty$,the $\frac{3}{x}$ tends to zero and the $\lfloor\frac{x}{4}\rfloor$ tends to $\infty$.I do not ...