Understanding the proof of limit points of $\{\frac{1}{m}+\frac{1}{n} :...
I am reading the solution of user Gregory Grant from this thread(Find the limit points of the set $\{ \frac{1}{n} +\frac{1}{m} \mid n , m = 1,2,3,\dots \}$) and I am having trouble following the...
View ArticleShowing that limit exists iff left and right limits are equal
Theorem of interest:The limit at an interior point of the domain of a function exists ifand only if the left-hand limit and the right-hand limit exist and areequal to each other.I'm using the $...
View ArticleProve that if the dot product is the product of the norms, then the vectors...
Let u, v be n-tuples in R^n. Recall the Cauchy-Schwarz Inequality: |<u,v>| <= |u||v|.Prove that |<u,v>| = |u||v| if and only if the u,v are linearly dependent, that is u = 0 or v = au...
View ArticleHelp sorting out a typo; outer-measure induced by $\rho((a,b))=(b-a)^2$ is zero.
I was wondering if anybody can help me figure out what the following exercise is supposed to say, because as it's stated I'm nearly certain this is wrong.Let $\mathcal{E}$ be the set of all open...
View ArticleWhat is this theorem and where can I find its detailed proof?
Here is the theorem I want to prove:I guess long ago when I studied analysis this theorem was given in some textbook with its proof. I think it was Kolomogorov book, or Royden Real analysis fourth...
View ArticleProve that the following limit is $1/2$
Let $(x_n)_{n\geq1}$ be a sequence of positive numbers such that$$x_{n+1}=x_n+x_n^{1/x_n}$$Prove that$$\lim_{n\to\infty}\frac{x_n-n}{(\log(x_n))^n}=1/2$$I tried using Stolz-Cesaro because the numerator...
View ArticleSum of two quasi-integrable functions
Let $f,g: X\to \overline{\mathbb{R}}=\mathbb{R}\cup \{\infty, -\infty\}$ be quasi-integrable functions w.r.t. the measure space $(X,\mathcal{M},m)$, i.e., $\int_X f= \int_X f^+-\int_X f^-$ and $\int_X...
View ArticleFinding maximum for a function of $2^b$ non-negative integer variables
Given, $M$ is a positive integer and,$$ 0 \le n_i \le M \hspace{10mm} \forall \hspace{2mm} 1 \le i \le 2^b $$ Then:$$\max\left\{\sum_{1 \leq i \le j \le 2^b} n_i n_j\right\} \hspace{2mm} \text{is}...
View ArticleExplanation Request - Stephen Abbott's Understanding Analysis proof that √2...
In Abbott's proof for existence of $\sqrt{2}$, he takes the following steps (which completely, utterly, totally baffle me):Choose 𝛼 as an upper bound for the set $T = \{t∈\mathbb R:t^2<2\}$ such...
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For $ p > 1$, show that:$$\lim_{n \to \infty} \left\{ \frac{n}{p-1} - \left[ \left(\frac{n}{n+1}\right)^p + \left(\frac{n}{n+2}\right)^p + \cdots\right] \right\}= \frac{1}{2}$$I could compute the...
View ArticleExistence of an injective continuous function $\Bbb R^2\to\Bbb R$?
Let's say $f(x,y)$ is a continuous function. $x$ and $y$ can be any real numbers. Can this function have one unique value for any two different pairs of variables? In other words can $f(a,b) \neq...
View ArticleHow to solve this exercise: prove that for every x, y from R that the...
How to solve this exercise:prove that for every x, y from R that the following property is satisfied: sum k = 1 to n - 1 [x + k/n] = [nx]
View ArticleProve that $f(x) = \sum_{n=1}^{\infty} \frac{sin^3(nx)}{n}$ is discontinuous...
Prove that $f(x) = \sum_{n=1}^{\infty} \frac{\sin^3(nx)}{n}$ is discontinuous at point $0$.I rewrote this function using $\sin^3(x) = \frac{1}{4} (3 \sin(x) - \sin(3 x))$.$f(x) = \frac{3}{4}(\sum...
View ArticleIf $limsup(S) = c$ then there exist finitely many $x \in S$ such that $x > c...
This question comes from Serge Lang's Undergraduate Analysis section II-1 problem 11.Let $S$ be a bounded set of real numbers. Let $A$ be the set of its points of accumulation. That is, $A$ consists of...
View ArticleEvaluate $\int_0^{\pi/2}...
The following problem is proposed by Cornel Ioan Valean, that is, to prove that\begin{align}& \int_0^{\pi/2} \operatorname{Li}_{2}^{2}\left(\sin^{2}\left(x\right)\right){\rm d}x=\int_0^{\pi/2}...
View ArticleProve that $f(x) = \sum_{n=1}^{\infty} \frac{\sin^3(nx)}{n}$ is discontinuous...
Prove that $f(x) = \sum_{n=1}^{\infty} \frac{\sin^3(nx)}{n}$ is discontinuous at point $0$.I rewrote this function using $\sin^3(x) = \frac{1}{4} (3 \sin(x) - \sin(3 x))$.$f(x) = \frac{3}{4}(\sum...
View ArticleShow that the $T$ is a compact operator
Let $\Omega\subset\mathbb R^2$ be bounded. Consider the partial differential equation $(P)$\begin{align}-\Delta u+u^5&=h \ \ \text{on }\Omega,\newline u&-0\ \ \text{on...
View ArticleIntegral $\int_{0}^{\pi/2} \arctan \left(2\tan^2 x\right) \mathrm{d}x$
The following integral may seem easy to evaluate ...$$\int_{0}^{\Large\frac{\pi}{2}} \arctan \left(2 \tan^2 x\right) \mathrm{d}x = \pi \arctan \left( \frac{1}{2} \right).$$Could you prove it?
View ArticleDifferentiability on an open interval around x0 implies that the derivative...
I am trying to prove to myself if the statement : “f’ defined on (a-R,a+R) is enough to say that f’ is then continuous at a”. I tried to write a proof about this because I have a strong feeling or...
View ArticleIs it true that the critical level set of an isolated critical point is a...
Equivalently, let $p$ be an isolated critical point of the smooth real-valued function $f$, is it true that there exists a contractible neighborhood $U$ of the point $p$ in $f^{-1}(f(p))$?
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