Closedness of sets
Let $A$ be a closed set in $\mathbb{R}^n$ such that $0\notin A$. Can we deduce that the set $B=\{\frac{x}{\|x\|}:\, x\in A\}$ is also closed? Thank you in advance.
View ArticleGiven a continuous function which it's improper integral converges,...
Prove/Disprove:$(1)$ If $f$ is a continuous function such that $f > 0, \forall x \in [0,\infty)$ and $\int_0^\infty f $ converges, then $\lim_{x\to \infty}f(x)=0.$$(2)$ If $f$ is a continuous and...
View ArticleHow should I should prove $\mathbb{R}\sim\{0,1\}^{ \mathbb{N}}$ [duplicate]
I've seen some argument about the binary representation, but I think it is not accurate because under some extreme cases, the rounding or bit constraint would results distinct reals also have the same...
View ArticleIs there an actual number next to a given number? [closed]
If the given number is for example 3/2, what number bigger than 3/2 is next to it?
View ArticleProve that $T_a M \subset ker(Df(a))$
I would like to show for an open subset $ U \subset \mathbb{R}^n$ and $f : U \to \mathbb{R}$ continuously differentiable and $M = \{x \in U | g_1(x) = 0, ..., g_r(x) = 0 \}$ with $r \leq n$. Let $a \in...
View ArticleBasis of alternating tensors
I am reading the book Calculus on Manifolds by Spivak and I am trying to solve problem 4.1(b):Let $e_1, \dots, e_n$ be the usual basis on $\mathbb{R}^n$ and let $\varphi_1, \dots, \varphi_n$ be the...
View ArticleMetric for connected path space.
I'm trying to prove the next function is a metric for the space of connected paths $T_{x,y}(X)$ where $x,y\in X\subset\mathbb{R}^{n}:$$$d(x,y)=\inf\{L(\sigma):\sigma\in T_{x,y}(X)\},$$ where...
View ArticleUniform Convergence of a Sequence of Differentiable Functions
I'm currently studying real analysis and I've come across a problem that I'm having trouble with. The problem is as follows:Let $(\phi_n)$ be a sequence of differentiable functions on $[a,b]$ such that...
View ArticleDetermining Absolute Convergence of a Series
Question:Hello, I'm currently studying series in my calculus course and I've come across a problem that I'm having trouble with. The problem is to determine whether the following series is absolutely...
View ArticleProving a Property of Convex Functions
Question:Hello, I'm currently studying convex functions and I came across the following property that I'm trying to prove:Suppose $g$ is convex on $\mathbb{R}$, show that for any...
View ArticleProving an Inequality for a Differentiable Function
Question:Hello everyone,I'm currently studying real analysis and I've come across a problem that I'm having trouble with. The problem is as follows:Suppose that $$g$$ is differentiable on $$(a,b)$$....
View ArticleProving that the set of half-integers is a model of the natural numbers using...
I postulate that the following set $\{0,0.5,1,1.5,...\}$ represents the natural numbers. Of course, intuitively, this isn't true. But let me try to show this using Peano's axioms. I'll first define the...
View ArticleCondition for existence of Fourier transform?
We can convert signal into frequency domain using Fourier transform. But I think we can't compute Fourier transform of any signal . Fourier transform also should have some limits.So I want to askis...
View ArticleConnected subsets of $\mathbb{Q}^{n}$ have at most one point
Given a set $A \subset \mathbb{Q}^{n}$ with #$A > 1$, is it possible to find $s \in \mathbb{R}^n$, $r \in \mathbb{R}_{> 0}$ such that $a_1 \in B(s,r), a_2 \in \operatorname{int}(\mathbb{R}^{n} -...
View ArticleAnother general form of beta function evaluation
Is there a general formula for this integral, $$\int_0^1 \frac{x^a (1-x)^b}{(c+hx)^{a+b+2}}dx$$I encountered an integral which broke into two smaller integrals and both of them were of the above form.I...
View ArticleConjecture about prime twins and definite integrals [closed]
If $n$ is such that $2nā1$ and $2n+1$ are twin primes,$$ā«_1^0\frac{\log(1+x^{2n+\sqrt{4n^2ā1})}}{1+x}dx$$appears to be$$\frac{\pi^{2}}{12}\left(n-\sqrt{4n^{2}-1}\right)+\log(a+b\sqrt{2n-1})\log(c+d...
View ArticleApproximating exponential function using piecewise constant function
I want to construct a piecewise function to approximate the function $f:[0,1]^d \to \mathbb{R}, ~f(x) = \exp(\|x\|^2)$. My approach is to partition the space $[0,1]^d$ into non-overlapping cubes...
View ArticleProve that $(a_n)$ and $(b_n)$ are converging sequences and whether $(c_n)$...
I'm solving some previous calculus exams, and this question came up:Let $(a_n)$ be a sequence that defined as $a_1=1/2$, $a_{n+1}=a_n^2+a_n^3$.Let $(b_n)$ be a sequence that defined as $b_1=-1$,...
View ArticleHow Determine the set of efficient extreme points using the parameterization?
I am writing to you because I need you to explain the solution to this exercise. I am very sorry, I have not found anything that will help me solve it. I know that in this forum there are teachers with...
View ArticleIf $\lim\sup a_n = -\infty$, then $a_n\to -\infty$, and if $\lim\sup a_n =...
Show that if $\lim\sup a_n = -\infty$, then $a_n\to -\infty$, and if $\lim\sup a_n = \infty$ there exists a subsequence of $a_n$ that $\to\infty$. What if $\lim\inf a_n = \pm\infty$? We're working with...
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