Let $u_1=\frac1N$, where $N\in\mathbb{Z^+}$, and $u_n=u_{n-1}+(u_{n-1})^2$....
I was playing with recursively defined sequences, and stumbled upon a curious apparent result.Let $u_1=\frac1N$, where $N\in\mathbb{Z^+}$, and $u_n=u_{n-1}+(u_{n-1})^2$.Let $f(N)=$ number of terms in...
View ArticleProving the solution of a functional equation is constant using Kronecker's...
I'm working on the following problem:Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function satisfying $f(x) + f(x + \sqrt{2}) = f(x + \sqrt{3})$ for all $x$. Prove that $f$ is constant.This...
View Articlecutoff function for fundamental solution
Consider the fundamental solution $h(x)=c_{n}|x|^{-(n-2)}$ defined for $x\in\mathbb{R}^{n}$, $x\neq 0$ with $c_{n}$ a normalizing constant. Is there some way to define a new function $H$ so that it is...
View ArticleOn the uniqueness of Caratheodory Extension Theorem when using semi-rings
Suppose that $X$ is a set, $J$ is a semi-ring with respect to $X$, $f$ is a pre-measure defined with respect to $J$, by Caratheodory's Extension Theorem there exist a measure $\mu$ such that $\mu$...
View ArticleTaylor Series expansion for a composition of functions
The Taylor Series expansion for $\frac{1}{1-x}$ is convergent for every real number $-1 < x < 1$.\begin{equation*}\frac{1}{1 - x} = \sum_{n=0}^{\infty} x^{n} .\end{equation*}Since $0 \leq x^{2}...
View ArticleProof that twice-differentiable function f on an interval I, $|h''(x)|\le M...
I'm currently working through Jay Cummings' Real Analysis textbook (self study), and have got stuck on a question on differentiation. The question is as follows:Suppose that $h:[0,5]->\mathbb{R}$ is...
View ArticleShowing the family $\{x^n/n \}_{n\in\mathbb{N}}$ is equicontinuous on $[0,1]$
As the title implies, let $\{x^n/n \}_n\subseteq C[0,1]$, where $x\in X$ and $X$ is a metric space. I want to show that this family is equicontinuous at each $x\in[0,1]$. By the definition then, we...
View ArticleA binary sequence such that sum of every 10 consecutive terms is divisible by...
Let $(a_n )_{n=1} ^{\infty} $ be a sequence of elements in $\{0, 1\}$such that for all positive integers $n,\Sigma_{i=n}^{ n+9}a_i$ is divisible by 3 . Then there exist a positive integer $k$ such that...
View Article$A+B$ closed if $A$ and $B$ are closed [duplicate]
Let $A,B$ be subset of $\mathbb R^n$, both closed.I want to prove that $A+B$ is closed too.I know how to prove with one of both sets is compact but not without.Please help me, i struggle finding a...
View ArticleDefinition of Bounded Sequence [closed]
What is the actual definition of Boundedness of a sequence? Like if I take the definition as $|a_n|\leq k$ for some $k>0$ then how could we possibly say every sequence is bounded in discrete metric...
View ArticleCan I simplify multiple inequality cases into a single absolute inequality?
I'm not an expert and I haven't dealt with these kinds of inequalities for a while, so I'm hoping someone can explain if my thinking process is reliable and will suffice to prove the results I'm...
View ArticleWeak convergence and approximations of the Dirac Delta [closed]
Suppose\begin{align*}a_n &\to \delta_0 \text{ as a distribution}, \\f_n &\stackrel{\ast}{\rightharpoonup} f \text{ in $L^\infty$} \\f_n \ast a_n &\stackrel{\ast}{\rightharpoonup} g \text{...
View ArticleElementary change of variable in an ODE that makes me doubt
I am quite ashamed of the level of my question, but I have this ODE -- it is a simplified version -- and my concern is only with the left hand side:$$ \dot v=kv$$which I need to rewrite in terms of c,...
View Article$\phi_{\sigma} (x) = O(|x|^{−α})$, as $x→±∞$.
I am currently studying a paper published in the "Journal of Approximation Theory" on Neural Networks. We have the following definitions:Definition(Sigmoidal functions):A function...
View Articlea function with every point a local maximum is constant on some subintervals...
I am working on the following problem:Let $f:\mathbb{R}\to\mathbb{R}$ satisfies that for all $x_0\in\mathbb{R}$, there exists some $\delta>0$ such that $f(x)\le f(x_0)$ for all $x\in (x_0-\delta,...
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Newton's method to solve for $f=ax^2-x+1=0$
Let $\{x_k\}$ be sequence generated by Newton's Method for solving$ax^2-x+1=0$, where $0<a\leq\frac{1}{4}$. Suppose $x_0<\frac{1}{2a}$.Show...
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 ArticleWhy do proofs using continuity use the Bolzano-Weierstrass theorem?
This may just be goofy of me but there's a lot of proofs I've been seeing recently while learning real analysis that keep on invoking the Bolzano-Weierstrass theorem, and while I understand how it's...
View ArticleWrite the set [0,1] as both a countable union and intersection of open intervals
I'm currently tackling the following question which has two parts:"Can you write the set $[0,1]$ as a countable intersection of open intervals. Either find suitable real numbers $a_1, a_2,....,b_1,...
View ArticleUnderstanding dense subset of $[0,1]$ with Lebesgue measure $\epsilon>0?$
When asked to find a dense subset of $[0,1]\subset\mathbb{R}$ with Lebesgue measure $\epsilon>0,$ there exist many solutions one may find; however, there is one which I have never understood...
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