Proof that the supremum of a continuous function is part of the range of that...
I'm following the textbook "Calculus" By Spivak. Currently, I'm reading a proof given for the following theorem:"If $f$ is continuous on $[a,b]$ then there exists an $x^*\in[a,b]$ such that $\forall...
View ArticleFinding a sequence of equations to equal another equation [closed]
I'm wondering what kind of problem space this might be?Lets say I have an equation eg $x + y = Q$Then I have a series of other equations involving $x$ and $y$. how might I determine what sequence of...
View ArticleI can prove the statement that a circle has definite indefinite [closed]
So let me first tell you that I have a circle, right? A circle definitely has a definite area. It doesn't have infinite area, right? Now, I can take a definite chord on managing the width of the chord,...
View ArticleConvergence in $L^{p}$ and convergence of derivative
Suppose $f_{n} \to 0$ strongly in $L^{p}_{loc}(\mathbb{R})$ for some $p \in [1, \infty]$, where $f_{n}$ is a smooth sequence. Is it true that $f_{n}' \rightharpoonup 0$ weakly in the sense of measures,...
View ArticleProve the sequence $\frac{1}{2}$, $\frac{1}{2+\frac{1}{2}}$,...
The recursion formula for the sequence is $a_{n+1}=\frac{1}{2+a_{n}}, n \ge 1, a_{1}=\frac{1}{2}$. The hint of the question was to prove {$a_{2n+1}$} is decreasing and bonded below, and {$a_{2n}$} is...
View ArticleWhy can we convert a power series of operators to a function, invert the...
I apologize as I am still somewhat unfamiliar with infinite series and knowing when and how we are allowed to use them, and especially with working with operators in this way. I imagine that I will be...
View ArticleWhat is $\prod\limits_{k=1}^{x}(a-1+k)$ when $x \in \mathbb{C} - \mathbb{N}$?
In my book Encyclopedia of mathematics and its applications 71 George E. Andrews, Richard Askey, Ranjan RoyIt is written that$$ \tag{1.1.1}x!=\frac{(x+n)!}{(x+1)_n}$$where $(a)_n$ denotes the shifted...
View ArticleExample of proof of an infimum?
I'm doing an excercise about infimums and supremums and I've seen different examples of proving a =inf(S) $\iff \forall \epsilon >0\exists s\in S\colon a+\epsilon>s$ and to me they just seem to...
View ArticleSchwartz theorem
Can i relax the conditions on f of the Schwartz theorem in real analysis?That is: if $f: \Omega \to R^2$, s.t $f \in C^2(\Omega)$, then i can change the order of derivatives and the result is the...
View ArticleWhat justifies the use of global coordinates when computing the...
Consider the $n$ dimensional torus $\mathbb{T}^n$. The $L^p$ spaces over $\mathbb{T}^n$ is defined as consisting of an equivalence class of functions satisfying:$$\int_{\mathbb{T}^n}|f|^p < \infty....
View ArticleIs it true that for any real valued function $f(t)$ on $\mathbb{R},$ $f(t)a +...
I have the following doubt.We know that for any two points $a$ and $b,$$L=ta+(1-t)b$ represents a line joining $a$ and $b.$ Is it true that for any real valued function $f(t)$ on $\mathbb{R},$$f(t)a +...
View ArticleWhat is meant by domain of function to be "explicitly defined"?
In my maths books it's said that: "when the domain of f(x) is not explicitly defined then in this case domain will mean the set of values of x for which f(x) assumes real value"SO what it meant by...
View ArticleHow do I test for convergence of $\sum\limits_{n = 2}^{\infty}...
I was trying to solve this problem.Test $\sum\limits_{n = 2}^{\infty} \frac{\log(n)}{n \sqrt{n + 1}}$ forconvergence or divergence .But I couldn't quite make a lot of progress. Here's what I tried.I...
View ArticleAn idea for this difficult integral:...
I am being stuck in caculating this integral: $$J=\int_{-\tfrac{1}{2}}^{\tfrac{1}{2}}\dfrac{\arccos x}{\sqrt{1-x^2}(1+e^{-x})}dx$$ I tried to change to another variable: $x = - t$ then $dx = - dt$,...
View ArticleConvolution preserve the Neumann boundary condition
Here, I want to know if convolution will preserve the Neumann condition or not. Suppose $K$ is a continuous function and integrable on some interval $[0,L]$, and $u$ is some 'good enouth' function that...
View ArticleTranscendental nature of natural log for proof validity?
I am following Understanding Analysis by Stephen Abott. I read a well bit into the book but I decided to go through and do the exercises through the book because I felt as if I wasn’t being rigorous.I...
View ArticleNotation Clarification for Taylor's Theorem in Higher Dimensions
The following lemma is stated in the coursebook for a paper I am taking on analysis in higher dimensions. (The Lemma is working towards proving Taylor's theorem in higher dimensions).I am having...
View ArticleWeak derivative of $|x|^{-\lambda}$.
We're supposed to show that for $\lambda\geq d-1$, there does not exist a weak derivative $v\in L^1_{loc}(\mathbb{R}^d)$ of $|x|^{-\lambda}$. Im really stuck here. Here's what I came up with so far:...
View ArticleHow to Prove Convergence of $\prod\limits_{n =1}^\infty...
For $\frac{\sin(\pi z)}{\pi z} =\prod\limits_{n =1}^\infty \left(1-\frac{z^2}{n^2}\right)$, prove convergence of $\prod\limits_{n =1}^\infty \left(1-\frac{z^2}{n^2}\right)$ for any $z \in...
View ArticleFollowing an answer given for exercise 6.9 in Rudin's RCA, convergence in...
In this link math.stackexchange.com/questions/522613, it is stated that the sequence of positive continuous functions $(g_n)$ converges towards $0$ in measure and that one can extract a subsequence...
View Article