If $\lim_{n \to \infty} n x_n = 0$ then $\lim_{n\to \infty} (1+x_n)^n =1$
The following appears without proof in some lecture notes about exponents (specifically, it was used to prove the limit $\lim_{n\to \infty} (1+\frac{x}{n})^n = e^x := \sup\{ e^q \mid q\in \Bbb Q, q\le...
View ArticleExtension of finitely additive function to a countably additive measure
Let X be an infinite set. Let $\mathcal{A}$ be the collection of subsets A of X such that either A is finite, and then m(A):=0, or the complement of A is finite, and then m(A):=1.Under what condition...
View ArticleBoundary properties of the convex envelope
Let$\Omega \subset \mathbb{R}^n$ be open, bounded and strictly convex, i.e. for all$x, y \in \overline{\Omega}$and$\lambda \in (0, 1)$we have that$$ \lambda x + (1 - \lambda)y \in \Omega.$$Furhermore,...
View ArticleReal Analysis , Folland Problem 6.1.5
Problem 6.1.5 - Suppose $0 < p < q < \infty$. Then $L^p \not\subset L^q$ if and only if $X$ contains sets of arbitrary small positive measure, and $L^q\not\subset L^p$ if and only if $X$...
View ArticleQuestion regarding Lebesgue Integrability in $\sigma$ -finite spaces
I'm taking a course in measure theory and we defined integrability in a $\sigma$ -finite space as follows: Suppose $\left(X,\mathcal{F},\mu\right)$ is a $\sigma$-finite measure space, a measurable...
View Article$\lim_{\epsilon \rightarrow 0} \frac 1{\epsilon} \int_0 ^{\infty} f(x...
Check if the following limit exist and find the value if it exist .$\lim_{\epsilon \rightarrow 0} \frac 1{\epsilon}\int_0 ^{\infty} e^{-x/{\epsilon} }(\cos (3x)+ x^2 +\sqrt{x+4} ) dx $My Attempt :-For...
View ArticleExample for a non-separable space of compactly supported continuous functions
For a topological space $X$ let $C_c(X)$ be the space of compactly supported continuous real-valued functions. It is well known that $C_c(X)$ (provided with the sup norm) is separable, if $X$ is a...
View ArticleIs Rudin's proof of the inverse function theorem correct?
This question concerns the proof of the inverse function theorem found in Walter Rudin's Principles of Mathematical Analysis (3rd ed.).It defines for each $y \in...
View ArticleDose $\lim_{k\to\infty}\langle f_k\rangle=\langle f\rangle$ imply...
We know that for any $\langle f\rangle\in L^p(X,\mathscr{A},\mu)$ there is a sequence $\{\langle f_k\rangle\}$ of elements of $L^p(X,\mathscr{A},\mu)$, where each $f_k$ is a simple function that...
View ArticleSubset of $\mathbb{R}$ that satisfies trichotomy and multiplicative closure...
Is it possible to construct a subset $\mathbb{P}$ of $\mathbb{R}$ thatsatisfies trichotomy and multiplicative closure but not additive closure?Trichotomy says that if $x\in\mathbb{R},$ then exactly one...
View ArticleShowing a $g\in L^{q}([0,1])$ and proving an inequality for its $q$-norm
I'm working through some problems dealing with $L^p$ spaces and came across this specific exercise:Let $1<p, q<\infty$ be such that $\frac{1}{p}+\frac{1}{q}=1$. For any $f \in L^{p}([0,1])$,...
View ArticleImage of an increasing function with discontinuities on a densely ordered...
When discussing this question, we made interesting observations on increasing functions with discontinuities on rationals. Let $f: [0,1] \to [0,1]$ be an increasing function with discontinuities on...
View ArticleFinding a fit function with specific properties
I am looking for a function f(x) for x>0 to fit to monte carlo simulation results. Here is an example of a simulated family of curves that I wish to fit:simulation resultsFrom the physical process...
View ArticleWell-posedness for system of linear PDES
I am studying systems of partial differential equations (PDEs) with smooth coefficients and am trying to understand the general solution. Consider the following system:\begin{align}u_x &=...
View ArticlePointwise convergence of almost periodic function
This question is inspired by this post and definitions I use can be found there.Fix $f$ a real-valued almost periodic (but not periodic) function. For each $t\in\mathbb{R}$, define $f_t(x) = f(x-t)$...
View Article$f \in C^{\infty} (\mathbb{R}^3, \mathbb{R}^4)$,...
We have $f \in C^{\infty} (\mathbb{R}^3, \mathbb{R}^4)$, $f = f_1, f_2, f_3, f_4$.Elements of that transformation satisfy:$$f_1(x)^2 + f_2(x)^2 = 1 = f_3(x)^2 + f_4(x)^2 \ \text{ for: } \ x \in...
View Article$\omega = \frac{x(x^2+y^2-1)}{(x^2+y^2+1)^2-4x^2}dy+...
I found such problem with solution.Problem:We have:$C1 = \{ (x, y) : (x-1)^2 + y^2 = \frac{1}{4} \}$$C2 = \{ (x, y) : x^2 +y^2 = 4 \}$$C3 = \{ (x, y) : (x + 1)^2 + y^2 = \frac{1}{4} \}$Those are...
View ArticleAbsolute convergence of series...
How do I prove that the following series converges absolutely$$\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}e^{-(2n+1)a}e^{(2n+1)m(b-a)}$$ if we know that $b<0<a$.The first thing I tried is proving that...
View ArticleDoes $\lim_{z \to z_0} f(z) < \infty$ imply that $f(z)$ is bounded near...
Does $\lim_{z \to z_0} f(z) < \infty$ imply that $f(z)$ is bounded near $z_0$? For example, do we have that since$$\lim_{z \to 0} \frac{z}{\sin z} = 1$$we have $\frac{z}{\sin z}$ is bounded near $z...
View ArticleDoes this limit exist, how to prove it exists and what is its value? [duplicate]
$$\lim_{n\to\infty}\frac1n\sum_{k=1}^n\frac{\sin(\frac{\pi k^2}{2n})}{\sin\frac{\pi k}{2n}}$$
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