# Tagged Questions

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### Integral Hölder bound

I was wondering if it is possible to find the following bound or if not, find a counterexample of it. Let $f\in C_0^1$ (compactly supported continously differentiable, in particular $\alpha$-Hölder ...
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### Strange inequality

I found the inequality $\beta e - \frac{3}{2} n \ log(e+Bn)+ \frac{5}{2} \ n \ log(n) + const \cdot n \geq \frac{\beta e}{2}+ \beta n$ in a textbook,provided that either $e$ or $n$ is large. We ...
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### Real Between Rationals

Let $x$ be a real number. Show that, for any $\varepsilon>0$, there exist two rationals $q$ and $q'$ such that $q<x<q'$ and $|q-q'|<\varepsilon$ How should I approach this prove?
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### How can I prove that $a^{2} < b^{2}$implies that $a < b$ in the Real Numbers? [on hold]

The answer to my question doesn't seem to exist elsewhere on the internet. I have the sets $A=\{ a : a\in R: a > 0,\ a^2 < 3\}$ and $B=\{ b: b\in R: b>0,\ b^2 > 3\}$, and I'm just ...
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### Limit superior does not increase when a sequence is replaced by its sequence of averages [duplicate]

Let $\{x_n\}$ be a bounded sequence of real numbers, and define a new sequence $\{\sigma_n\}$ by $$\sigma_n=\frac 1n\sum_{i=1}^nx_i.$$ Prove that $\limsup \sigma_n\le \limsup x_n$. I am confused on ...
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### Putting a bound on some probability inequality

Assume that we have the following polynomial: $$ax^2 + bx =c$$ and a, b, c are i.i.d uniform random variables in [0, 1]. I'm trying to calculate the probability that the root is real, and that ...
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### Using mean value theorem to show that $\cos (x)>1-x^2/2$

I have a question, by applying the mean value theorem to $f(x)=\frac{x^2}{2}+\cos (x)$, on the interval [0,x], show that $\cos (x)>1-\frac{x^2}{2}$. We know that ...
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### Help with a simple function inequality

Let $R\geq 1$, $f(x)$ a function and $1_A$ the indicator function of the set $A$; is this inequality true? \frac{\vert f(x)\vert}{R}1_{R\leq\vert x\vert\leq 2R}\leq\frac{\vert f(x)\vert}{1+\vert ...
I'm having some trouble justifying some steps in a paper. Let $a_n$ be an increasing sequence of integers satisfying $n! \le a_n \le 2(n!)$, and let $f:\mathbb{N} \to \mathbb{N}$ be a function ...
### How prove $\frac{x^{3}+y}{y^{3}+x}-1\geq \ln \frac{(x^{2}+1)^{2}}{x}-\ln \frac{(y^{2}+1)^{2}}{y}$?
How prove $\frac{x^{3}+y}{y^{3}+x}-1\geq \ln \frac{(x^{2}+1)^{2}}{x}-\ln \frac{(y^{2}+1)^{2}}{y}$ where $x, y\geq 1$?