Does there exist a positive sequence with these properties?
Let $\{x\}$ denote the fractional part of $x$ and $d_n=\text{lcm}(1,2,...,n)$. Does there exist a sequence with all positive terms $(a_n)_{n\geq 1}$ such that $$\lim_{n\to\infty} \{a_n\}=1\ \text{,}\...
View ArticleIs this step in the asymptotic analysis of a certain physical system ok?
In analyzing the thermodynamic limit of a certain system in statistical mechanics, I've encountered the following situation: $f_n$ is a sequence of probability density functions on the real line with...
View ArticleBaby Rudin: Corollary to Theorem 2.12 (at most countable union of at most...
I am trying to prove the corollary to Theorem 2.12 in Rudin's book. Theorem 2.12 says:Let $\{E_n\}$, $n = 1,2,3, \ldots$ be a sequence of countable sets, and put $S = \bigcup\limits_{n=1}^{\infty}...
View Articlehow can I show that $\bigcap\bigcup_{k=0}^{n-1} [{k \over n}, {k+1 \over...
Let $X=[0,1]$ given $\Delta=\{(x,x):x\in X\}$\begin{equation}\Delta_n = \bigcup_{k=0}^{n-1} \left[{k \over n}, {k+1 \over n} \right]^2 \end{equation}How can I show that $\Delta=\cap\Delta_n$? It's ok...
View ArticlePugh Analysis Integral Question
This is Exercise 39 of Chapter 4, Real Mathematical Analysis by Pugh.Let f : [0, 2π] → R be a continuous function such that$$\int_0^{2 \pi} f(x) \sin (n x) d x=0$$Then show that f is constant.I know...
View ArticleMax conditional entropy
Random vectors $(X,Y)$ are distributed over $\mathbb{R}^n \times \mathbb{R}^m$ with zero mean and covariance $\Sigma \in \mathbb{R}^{(n+m)\times (n+m)}_+.$What distributions attain the maximum value...
View ArticleBump function with integral $1$ and value $1$ at zero
How can i contruct a smooth bump function $F$ on $\Bbb{R}^n$ such that $F(0)=1$ and with integral $1$?I have tried to manipulate the function $f(x)=e^{-\frac{1}{x^2}}$ if $x>0$ and $f(x)=0$ if $x...
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M is composed of line segments connecting ellipse to $(0,0,0)$ Calculate...
I found such an exercise among my set of exercises preparing for exams and I have no idea how to solve that.Every point of ellipse $\{ (x,y,z) \in \mathbb{R}^3 : x^2 + y^2 = 1, x + z = 1 \}$ is...
View ArticleProve that $\mathbb{R}^m$ with Euclidean metric is complete
Prove that $\mathbb{R}^m$ with Euclidean metric is complete.Every Cauchy sequence in $\mathbb{R}$ is bounded and has monotonic subsequence thus has convergent subsequence. Since Cauchy sequence which...
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$z = xy$ intersects with $y = 2x^2$ on a curve. Points on the curve are...
A surface in $\mathbb{R}^3$ with equation $z = xy$ intersects a surface with equation $y = 2x^2$ along some curve. Each point of the arc of this curve contained in the area $0 < z < 1$ is...
View ArticleWeierstrass Approximation Theorem variant proof [closed]
I need to prove the following:For every $f \in C[0,1]$ there exists a sequence $(p_n)$ of polynomials such that$$\lim_{n \to \infty} \sup_{t \in [0,1]} |f(t) - p_n(t)| = 0.$$ Can you please direct me...
View ArticleDecomposition of $\Bbb R^n$ as union of countable disjoint closed balls and a...
This is a problem in Frank Jones's Lebesgue integration on Euclidean space (p.57), $$\mathbb{R}^n = N \cup \bigcup_{k=1}^\infty \overline{B}_k$$where $\lambda(N)=0$, and the closed balls are...
View ArticleShow every bounded infinite set has a maximum limit point and a minimum limit...
Show every bounded infinite set has a maximum limit point and a minimum limit point.Here is my thought even if it is not formalLet $S$ be bounded and infinite set.Bolzano–Weierstrass theorem: Every...
View ArticleLaplace transform as a linear transformation
The Laplace transform is a linear mapping from the functions of time domain to frequency domain. The part $K(t,p)=e^{-pt}$ is called the kernel. The definition of kernel is the set which takes the...
View Article$L^1$ and $L^\infty$
Let X be a subspace of Banach space $L^1 ([0,1])$, and for any element in X belongs to $L^\infty ([0,1])$.We set linear operator$A : (X,\|\cdot\|_1) \to (L^\infty ([0,1]), \|\cdot\|_\infty)$ as A(f) =...
View ArticleAn inequality involving $|(1/\zeta)^{(n)}(x)|$
I was comparing various functions with $|(1/\zeta)^{(n)}(x)|$ on Desmos when $x \geq 2$ the following inequality seems to be true for all values of $n$:$|(1/\zeta)^{(n)}(x)| \leq n!/(x-1)^n$Is there...
View ArticleProve the unbound sequence $\left\{{a_n} \right\}$ has a subsequence...
Let $\left\{{a_n} \right\}$ be an unbound sequence. Prove that thereexists a subsequence $\left\{{a_{p_n}} \right\}$ of $\left\{{a_n}\right\}$ such that $\left\{ \frac{1}{a_{p_n}} \right\}$ converges...
View ArticleApproximating powers of elements on the unit circle
Let $R \subseteq \mathbb{N}$. We say that $R$ is adequate if:$$\forall n \in \mathbb{N} \; \; \forall \varepsilon > 0 \; \; \forall w \in S^1 \; \; \exists r\in R \; \; |w^n - w^r| <...
View ArticleSaying solution of a PDE is continuous to the boundary
Suppose $\Omega \subset \mathbb{R}^n$ is a domain and $u$ is a function on it such that $\Delta u$ converges to $0$ at every boundary point of $\Omega$, can we say from here that $u$ extends...
View ArticleProof of absolutely convergent sums over two indices.
In the book Concrete Mathematics (2nd) written by Ronald Graham, Donald Knuth and Oren Patashnik, they prove the next theorem.Absolutely convergent sums over two or more indices can always be summed...
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