Intereting Posts

If $A$ is compact and $B$ is closed, show $d(A,B)$ is achieved
Compute $\int_0^1 \frac{\arcsin(x)}{x}dx$
Prove that the integral of an even function is odd
A counterexample
If $a$ and $b$ commute and $\text{gcd}\left(\text{ord}(a),\text{ord}(b)\right)=1$, then $\text{ord}(ab)=\text{ord}(a)\text{ord}(b)$.
Proof of implication: $\varphi^*\text{ is bounded below}\implies\varphi\text{ is a quotient map}$
What is the limit distance to the base function if offset curve is a function too?
Finding a polynomial with a given shape
Prove by induction that $n^3 + 11n$ is divisible by $6$ for every positive integer $n$.
The relationship between eigenvalues of matrices $XY$ and $YX$
Proof of $\gcd(a,b)=ax+by$
How to evaluate this limit: $\lim_{x\to 0}\frac{\sqrt{x+1}-1}{x} = \frac12$?
Proving limits using epsilon definition
Given complex $|z_{1}| = 2\;\;,|z_{2}| = 3\;\;, |z_{3}| = 4\;\;$ : when and what is $\max$ of $|z_{1}-z_{2}|^2+|z_{2}-z_{3}|^2+|z_{3}-z_{1}|^2$
Reciprocity Law of the Gaussian (or $q$-Binomial) Coefficient

I tried to prove something but I could not, I don’t know if it’s true or not, but I did not found a counterexample.

Let $(a_n)$ be a sequence in a general metric space such that for any fixed $k \in \mathbb N$, we have $|a_{kn} – a_n| \to 0$. Is $a_n$ necessarily a Cauchy sequence?

- Show $\int_0^\infty \frac{\cos a x-\cos b x}{\sinh \beta x}\frac{dx}{x}=\log\big( \frac{\cosh \frac{b\pi}{2 \beta}}{\cosh \frac{a\pi}{2\beta}}\big)$
- Evaluating $\int_0^\infty\frac{\log^{10} x}{1 +x^3}dx$
- Work to pump water from a cylindrical tank
- Closed-forms of the integrals $\int_0^1 K(\sqrt{k})^2 \, dk$, $\int_0^1 E(\sqrt{k})^2 \, dk$ and $\int_0^1 K(\sqrt{k}) E(\sqrt{k}) \, dk$
- Derivative of a factorial
- Rectifiability of a curve

- Partial fraction integration
- Geometric Interpretation of Total Derivative?
- Finding the catenary curve with given arclength through two given points
- How can we determine the closed form for $\int_{0}^{\infty}{\ln(e^x-1)\over e^x+1}\mathrm dx?$
- A limit question (JEE $2014$)
- A definition of Conway base-$13$ function
- For which $\mathcal{F} \subset C$ does there exist a sequence converging pointwise to the supremum?
- Prove convergence of hyperbolic recursive series
- How to find the center of mass of half a torus?
- suggest textbook on calculus

The additional question about adding the condition that the sequence is bounded caught my attention. (It is in Daniel’s comment under Brian’s answer.)

First I was able to get an example only for $k=2$:

$a_{2^x+y}=\frac y{2^x}$ for $0\le y<2^x$.

When trying to find an example working for all $k$’s I got stuck for some time, so I tried to modify Brian’s example:

$a_n=\sin\left(\frac\pi2 \lg\lg n\right)$.

To show that it fulfills the requirements, we can use $|\sin x-\sin y|\le |x-y|$ and the same reasoning as Brian did. (In detail: $|a_{kn}-a_n| = |\sin(\frac\pi2\lg\lg kn)-\sin(\frac\pi2\lg\lg n)| \le \frac\pi2(\lg\lg kn – \lg\lg n)$, an the RHS converges to 0, as shown in Brian’s answer. Thus $|a_{kn}-a_n|\to 0$.)

But this sequence has a subsequence convergent to 0 (for $n=2^{2^{2k}}$) and a subsequence convergent to 1 (for $n=2^{2^{4k+1}}$). Hence it is not convergent and, consequently, not Cauchy.

I hope I did not miss some mistake there.

It need not be a Cauchy sequence; here’s a counterexample in $\mathbb{R}$.

Let $a_n = \ln\ln n$. Then $|a_{kn} – a_n| = \ln\ln kn – \ln\ln n = \ln\frac{\ln kn}{\ln n} = \ln\frac{\ln k + \ln n}{\ln n} = \ln\left(1 + \frac{\ln k}{\ln n}\right) \to 0$ as $n \to \infty$, but the sequence $\langle a_n \rangle_n$ is unbounded and hence not Cauchy.

- Sanity check on example 6.5 from “Counterexamples in probability and real analysis” by Wise and Hall
- Round-robin party presents (or: Graeco-Latin square with additional cycle property)
- eigenvalues of certain block matrices
- Limit $\lim_{x\to 0} \frac{\tan ^3 x – \sin ^3 x}{x^5}$ without l'Hôpital's rule.
- A general element of U(2)
- How to prove that homomorphism from field to ring is injective or zero?
- Use Fourier series for computing $\sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}$
- subtraction of two irrational numbers to get a rational
- Find the eigenvalues of a matrix with ones in the diagonal, and all the other elements equal
- Find solution of equation $(z+1)^5=z^5$
- The proof of $e^x \leq x + e^{x^2}$
- Riemann integral proof $\int^b_a f(x) \, dx>0$
- Integrals of $\sqrt{x+\sqrt{\phantom|\dots+\sqrt{x+1}}}$ in elementary functions
- Consider the “infinite broom”
- Probability that n points on a circle are in one semicircle