Intereting Posts

No. of possible dense subsets of a metric space
Proving rigorously a map preserves orientation
Prove that $\lfloor 2x \rfloor + \lfloor 2y \rfloor \geq \lfloor x \rfloor + \lfloor y \rfloor + \lfloor x+y \rfloor$ for all real $x$ and $y$.
How to prove $\frac{1}{x}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+2\sqrt{\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}}$
Gaussian-Like integral
Proof of Ptolemy's inequality?
Sum of real powers: $\sum_{i=1}^{N}{x_i^{\beta}} \leq \left(\sum_{i=1}^{N}{x_i}\right)^{\beta}$
The non-existence of non-principal ultrafilters in ZF
Gauss sums and module endomorphisms
Upon multiplying $\cos(20^\circ)\cos(40^\circ)\cos(80^\circ)$ by the sine of a certain angle, it gets reduced. What is that angle?
Contribution (weighted average) of change in rate over time
Is $\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu$ a convergent integral?(2)
$A$ is an affine $K$-algebra and $f$ a non-zero divisor of $A$. Can one say that $\dim A=\dim A_f$?
convergence/or divergence of the series $\sum_{n=1}^{\infty}(a_n+b_n)$ given the convergence/divergence of component series
How to derive the Golden mean by using properties of Gamma function?

This is a more specific variation of the question in the post

Existence of a sequence with prescribed limit and satisfying a certain inequality

Suppose you have two infinite sequences $\{a_n\},\{b_n\}$, with $0<a_n<b_n < 1$ for each $n$, such that both $a_n, b_n \to 1$ as $n \to \infty$. **Further assume that $a_n/b_n \to 1$** as $n \to \infty$.

- Elementary proof for $\lim_{n \to\infty}\dfrac{n!e^n}{n^n} = +\infty$
- How to evaluate the integral$ \frac{\log x}{1+x^2}$
- What is “advanced calculus”?
- Is there a function with a removable discontinuity at every point?
- Completing the differential equation from exercise 10.23 in Tom Apostol's “Mathematical Analysis”
- Limit of $f(x+\sqrt x)-f(x)$ as $x \to\infty$ if $|f'(x)|\le 1/x$ for $x>1$

Does there exist a sequence $\{s_n\}$ with $s_n \to 1$ as $n \to \infty$ such that

$$

(1 – a_n) \leq (1-b_n)s_n

$$ for $n$ large enough?

The counterexamples given in the post above used sequences $a_n, b_n$ which grew asymptotically different (but which had the same limit). This question forces that $a_n \sim b_n$.

- Show that $\lim\limits_{x\to0}\frac{e^x-1}{x}=1$
- How to obtain asymptotes of this curve?
- Why is $\int^\infty_{-\infty} \frac{x}{x^2+1} dx$ not zero?
- Chernoff Bounds. Solve the probability
- Convergence of a recursively defined sequence
- A definite integral with trigonometric functions: $\int_{0}^{\pi/2} x^{2} \sqrt{\tan x} \sin(2x) \, \mathrm{d}x$
- Methods for choosing $u$ and $dv$ when integrating by parts?
- Infinite Product $\prod_{n=1}^\infty\left(1+\frac1{\pi^2n^2}\right)$
- Is this a correct use of the squeeze theorem?
- Multivariable Epsilon-Delta Proof?

- Convergence of sequence in uniform and box topologies
- Connectedness problem: sequences of points with distances at most $\varepsilon$
- find area of the region $x=a\cos^3\theta$ $y=a\sin^3\theta$
- Is this AM/GM refinement correct or not?
- Why isn't there a fixed procedure to find the integral of a function?
- Prove that $x^4+x^3+x^2+x+1 \mid x^{4n}+x^{3n}+x^{2n}+x^n+1$
- Are there open problems in Linear Algebra?
- expected value of a sum of a 10 sided die
- Why is this sequence of functions not uniformly convergent?
- “Random” generation of rotation matrices
- Bijection abstract simplicial complex
- Computing the product of p/(p – 2) over the odd primes
- Extension of continuous function on a closed subset of $\mathbb{R}$
- On the eigenvalues of “almost” complete graph ?!
- why don't we define vector multiplication component-wise?