Metric space property of balls
Is it true that for a complete metric space $X$, for every continuous function $\epsilon:X\rightarrow (0,\infty)$, there exists a continuous function $\delta:X\rightarrow (0,\infty)$ such that for any...
View ArticleDoes this prove that $\lim_{x \rightarrow \infty} \frac {P(x)} {{Q}(\log x)}...
I am trying to show that $\lim_{x \rightarrow \infty} \frac {P(x)} {{Q}(\log x)} = \infty$, where $\lim_{x \rightarrow \infty} f(x) = \infty \iff \forall N \in \mathbb{R} \exists x_n \in \mathbb{R}^{+}...
View ArticleLimit of Function Guaranteed by Sequence Properties
Let $S\subset\mathbb{R}$, and $c$ a cluster point of $S$.Define $f:S\to\mathbb{R}$, and suppose that for every sequence ${(x_n)_{n=1}^{\infty}}$ in $S\setminus \{c\}$ such that...
View Article$f(I)$ is a graph of a differentiable function (I'm almost there)
Let $f:I\to \mathbb R^2$, defined by $f(t)=(x(t),y(t))$ a differentiable path. I would like to prove the following implication$$\text{If $x'(t)\neq 0$ for every $t\in I$, then $f(I)$ is a graph of a...
View ArticleSome doubts about the series of sets [closed]
Why does the sum of countable numbers, if conditionally convergent, change its limit after rearranging those numbers, whereas the union of countable sets does not affect the limit when the order is...
View ArticleIs every closed (or open) ball in a normed vector space convex?
I am solving a problem and I need to use this fact:Every closed ball (or open ball) in the Euclidean space $\Bbb R^n$ is convex.Is it true?
View ArticleLimit of function with different speeds of divergence
Can we say $\lim_{x \to 0} f(x) = \infty$, where $f(x)$ is defined as, for example :$$ f(x) := \begin{cases} \frac{1}{x} & x > 0 \\ - \frac{2}{x} & x < 0 \end{cases} $$Both limits $x \to...
View ArticleDominating function for $\lim_{n \rightarrow \infty} \int_0^1...
Compute $\lim_{n \rightarrow \infty} \int_0^1 \frac{n+n^kx^k}{(1+\sqrt{x})^n}$, where $k \geq 1$.Letting $f_n(x) := \frac{n+n^kx^k}{(1+\sqrt{x})^n} $, we have the pointwise limit $f_n \rightarrow 0$...
View ArticleIf $f: U \to \mathbb{R}^3, U \subset \mathbb{R}^4$ an open set, is...
I study mathematics as a hobby and therefore do not have access to a professor to check if my work is correct, so I am coming here for help. Can you please check if my answer is correct?The exercise is...
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In proving completeness of $\ell^1$, how do we show that the real sequences...
I have two doubts on the initial statements on this proof of the completeness of $l^1$.I don't understand why we can say that the sequence $x_k^{(n)}$ is a Cauchy sequence in $R$. I understand that...
View ArticleQuestion About The Remark after Proposition 1.4.11 from Measure Theory by...
My QuestionDefine subsets $G$, $G_0$, and $G_1$ of $\mathbb{R}$ by\begin{align*} G &= \{x:x=r+n\sqrt{2}\ \text{for some $r$ in $\mathbb{Q}$ and $n$ in $\mathbb{Z}$}\},\\ G_0 &=...
View ArticleGiven a closed set in a preordered metric space. Does its minimum value...
Let $(X,d)$ be a metric space, and also $(X,\le)$ is a preordered set. If given a subset $E\subset X$ and $E$ is a closed set, can we always find the minimum value $\beta$ of $E$, such that $\forall...
View ArticleIs the Schwartz semi-norm a norm?
In the script for my lecture, there is an exercise asking for a proof that$$\|\varphi\|_{\alpha, \beta} = \sup_{x \in \mathbb{R}^d} |x^{\alpha} \partial^{\beta}\varphi(x)|$$is not a norm on...
View ArticleA Notion of Continuity Based on Existence of Certain Sequences
I am interested in the name for the following notion of continuity:Say that the function $f:\mathbb{R}^d \to \mathbb{R}$ is $(??)-$continuous at $x$ if there exists a sequence $x_n\to x$ with $x_n\neq...
View ArticleProof of continuity for a generalized form of the Dirchlet function?
Let $f$ be a real-valued function defined by:\begin{equation}f(x)= \begin{cases} f_1(x) & \text{if } x \in \mathbb{Q}\\ f_2(x) & \text{if } x \in \mathbb{Q}^c \end{cases}\end{equation}Show that...
View ArticleIs Lebesgue outer measure semicontinous?
Let $m^*$ be the Lebesgue outer measure on $\mathbb{R}$. Endow $2^{[0,1]}$ with the product topology, which is Hausdorff compact. Identify it with the set of all subsets of $[0,1]$. Is the map...
View ArticleBounded variation + injective implies piecewise strictly monotonic almost...
The question is can we prove the following: Consider $a,b\in {\bf R}$, $a<b$.(1) If $u\in {\rm BV}([a,b])$, and(2) if $u$ is an injection on $[a,b]$,then there exists $u_0:[a,b]\rightarrow {\bf R}$...
View ArticleInterchangability of expected value and integral only when $\mathbb E|X(t)| <...
In my textbook of Stochastic Process (in Chinese), I found a theorem as follows:Theorem: If a stochastic process $\{ X(t), t \ge 0 \}$ has $\mathbb E|X(t)| < \infty$ and $\mathrm{var} X(t) <...
View ArticleSmallest integer $n$ such that $\frac{x^n}{(n+1)!}
Consider the following setup:Let $x, c \in \mathbb{R}_{> 0} $ be fixed.Is there a "simple" formula, for the smallest integer $n$ that satisfies $\frac{x^{n}}{(n + 1)!} < c$?The inequality looks...
View ArticleIs there an ODE so that every solution is extensible?
The following question arose while studying initial value problems ($y(t_o)=y_0, y'(t)=f(t,y(t)))$:An solution to an IVP is a function $u:J\to \Omega$, where $J$ is an open interval and...
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