What does it mean to show that an integral exists?
QuestionI am slightly unclear on some terminology that I have come across in the following question:Let $f(x, t) = xe^{−xt}$.Show that the integral $I(x) = \int_0^{\infty} f(x, t)\ dt $exists for all...
View ArticleSign permanence of locally Lipschitz functions calculated on a sequence
Suppose I have a sequence $a_k(m)>0$ with $m\in\mathbb{N}$ such that, given $k\in\mathbb{N}$ and $p\geq 1$, I can show that $$|a_{k+p}(m)-a_k(m) |\leq \frac{m^2}{k^2}$$ with $a_{k+p}<a_{k}$. I...
View ArticleProve convergence of $\limsup_{n\to\infty}$
I am new to Real Analysis, and I have found this problem hard to formalize.ProblemLet $(p_n)_{n\in\mathbb{N}}$ and $(q_n)_{n\in\mathbb{N}}$ sequences such that $(p_n)\to u$ and $(q_n)\to v$. Consider...
View ArticleDefinition of class of continuously differentiable functions on a closed...
This is a technical simple question. The space $C^1([a,b])$ is defined as$$C^1([a,b])=\{f:[a,b]\to R: \exists\; f' \textrm{ and is continuous on } [a,b] \}.$$I think there is some explaining to do...
View ArticleProb. 12, Chap. 4, in Baby Rudin: A uniformly continuous function of a...
Here is Prob. 12, Chap. 4, in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:A uniformly continuous function of a uniformly continuous function is uniformly continuous.State...
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A function whose points of discontinuity form an arbitrarygiven $F_\sigma$ set.
From 'Counterexamples in Analysis' by Bernard R. GelbaumJohn ,M. H. Olmsted i studied this:The part underlined in red is unclear to me.If $c\in (A_n\setminus A_{n-1})\setminus I(A_n\setminus A_{n-1})$...
View ArticleHow to write this in a rigorous way? [duplicate]
I was studying for a test and encountered the following problem in a textbook:Suppose $f(x)$ is continuous, with $f >0$ for all $x$ and $\lim_{x \to \infty} f(x) = \lim_{x \to -\infty}f(x)= 0$Show...
View ArticleProve/disprove that $a^{2m} + b^{2m} + c^{2m} > 2^{1-m}$ subject to $a + b +...
Problem. Let $a, b, c$ be reals with $abc\ne 0$, $a + b + c = 0$, and $a^2 + b^2 + c^2 = 1$. Prove or disprove that$a^{2m} + b^{2m} + c^{2m} > 2^{1-m}, \forall m\in \mathbb{Z}_{>2}$.Prior...
View ArticleSequence of measurable functions converging a.e. to a measurable function?
I understand if $(X, \Sigma, \mu)$ is a measure space, and we have a sequence of measurable functions $f_{n}$ such that $\lim \limits_{n \to \infty} f_{n}$ exists almost everywhere d$\mu$ (a.e....
View ArticleA simple clarification on convergence of functions
Definition:$\lim_\limits{\large x\ \to\ x_0 \atop \large x\ \in\ E}f(x) = L$ iff for every $\epsilon > 0$, there exists a $\delta > 0$ such that $\vert f(x) - L \vert \leq \epsilon$ for all $x...
View ArticleContinuity of family of 1-parameter maps in Milnor's Morse theory
In the proof of Theorem 16.3 in Milnor's Morse Theory (see. pp. 91), he writes "It is easily verified that $r_u(\omega)$ is continuous as a function of both variables." I'm struggling to see why this...
View ArticleWhat's the nth term of the sequence 2,5,10,17,26 [closed]
What's the nth term of the sequence 2 5 10 17 26
View ArticleHow to determine whether $\sum_{n=1}^{\infty}\frac{1}{n^x}$ converges...
I have learned some theorems about how to determine the convergence of a series of functions, such as the continuity, integrability, differentiability theorems and Dini's theorem, as well as the...
View ArticleClosed form for this binomial double sum?
I've found this sum:$$S(n) = \sum_{k=n}^{2n-1} k(-1)^k \sum_{i=k}^{2n-1} {2n \choose i+1} {i \choose n}$$The inner sum is elliptical iirc, but perhaps the double sum has a nice expression. We can...
View ArticleProof that the Hamming distance is a metric [duplicate]
If $x$, $y$, are words of length $n$ in a code $C$, $x=x_1 \cdots x_n$, $y=y_1 \cdots y_n$ we define$$d(x, y)= d(x_1, y_1) + \cdots + d(x_n, y_n)$$from where$$\quad d(x_i, y_i)=0 \text{ if } x_i=y_i...
View ArticleHow does Axler know he has found the infimum?
I am reading the following example from Measure, Integration & Real Analysis by Sheldon Axler about the outer measure:Suppose $\displaystyle A=\{ a_1,a_2,...,a_n \}$ is a finite set ofreal numbers....
View ArticleIs it possible to show this Integral identity, by assuming the hyposeses I...
Assume there exists $p>2$ such that $B\in L^p_{\rm loc}(\mathbb{R}^2)$. Assume in addition that there exists $\tau >2$ such that\begin{equation} \label{B-decay-cond}|B(x)| \, = \, O(|x|^{-\tau})...
View ArticleComputing the integral $ \int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2}...
Integrate$$\int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2} \ln(1-2x\cos\phi+x^2) \, d\phi.$$Something that may help $(1-2x\cos\phi+x^2)=(1-xe^{i\phi})(1-xe^{-i\phi})$. And using the series...
View ArticleProve that $c \sup A = \sup(cA)$ for $c>0$.
I'm new to real analysis and trying to prove $\sup(cA)=c\sup(A)$ for $c>0$. Using this definition of least upper bound:$s=\sup A$, where $s\in \mathbb R$ and $A\subseteq \mathbb R$ if$\forall a\in...
View ArticleUniformly Continuous and locally Lipschitz but not Globally Lipschitz...
I know such a function exists but I can’t find an example. I have the famous example f:]0,inf[ --->R f(x)=sqrt(x) function. But I can't find any other function which is Uniformly Continuous and...
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