Articles of analysis

Convergence and uniform convergence of a sequence of functions

Suppose that ($f_n$;$n \geq 1$) is a sequence of functions defined on an interval [a,b].We will say $f_n$ tends to $f$ uniformly on $[a,b]$ as n tends to infinity if for every $\epsilon>0$ there exists a $N$ so that; $n \geq N$ implies $|f_n(x)-f(x)|<\epsilon$ for all $x \in [a,b]$. $(i)$ Consider the functions $f_n:[0,2] \rightarrow […]

Is $f(x)=\sup_{y\in K}g(x, y)$ a continuous function?

Let $K\subset \mathbb R^n$ be a compact subset and consider a continuous function $g:K\times K\longrightarrow \mathbb R$. Define $f:K\longrightarrow \mathbb R$ by, $$f(x)=\sup_{y\in K}g(x, y).$$ Is $f$ a continuous function?

Precise definition of epsilon-ball

My textbook gives the following definition: “For each $\epsilon>0$, the $\epsilon$-ball about a point $x$ in a metric space $M$ is the set $\{y\in M:d(x,y)<\epsilon\}$.” Is this correct? Because this sounds as if $\epsilon$ is just a dummy variable, and that there is such a thing as, say, a “5-ball” meaning $\{y\in M:d(x,y)<5\}$. Shouldn’t the […]

Problems with a proof that every real sequence has a monotonic subsequence

I have some problems with the visualization of this proof. (I will present the problems I have with it in the end, with some intuitive thoughts related to them in italics.) Theorem: Every real sequence has a monotonic subsequence. Proof (Thurston): Take any $(x_m) \in R^\infty$ and define $S_m := \{x_m, x_{m+1},\dots \}$ for each […]

About measure theoretic interior and boundary

Reference:- Evans-Gariepy, Federer, other books and notes of geometric measure thoery. I just want to clarify whether these definitions of measure theoretic interior and boundary are correct. Given a Borel regular measure $ \mu $ in $\mathbb{R}^n $, given a $\mu$-measurable subset $E \subset \mathbb{R}^n $, let $$ \psi(x,E) = \lim_{r\rightarrow 0}\frac{\mu(E\cap B(x,r))}{\mu(B(x,r))} $$ Here […]

What is the expansion in power series of ${x \over \sin x}$

How can I expand in power series the following function: $$ {x \over \sin x} $$ ? I know that: $$ \sin x = x – {x^3 \over 3!} + {x^5 \over 5!} – \ldots, $$ but a direct substitution does not give me a hint about how to continue.

If the condition of differentiability holds for the rationals then the function is differentiable?

$f:\mathbb{R}\rightarrow\mathbb{R}$ continuous, $a\in\mathbb{R}$. Suppose that there exists $L\in\mathbb{R}$ such that for every $\varepsilon>0$ there exists $r(\varepsilon)>0$ such that $|\frac{f(x)-f(a)}{x-a}-L|<\varepsilon$ for every $x\in\mathbb{Q}$ and $|x-a|<r(\varepsilon)$. I have to show that $f$ is differentiable at $a$ with $f^\prime(a)=L$. I know that this should be a trick using the density of the rational, but I can’t see it, […]

Supremum of two subtracted fractions less than one

Let $$S=\left\{\frac{1}{n}-\frac{1}{m}: m,n∈\mathbb{N}\right\}$$ Find the Supremum of this set. I get the feeling that the answer is $1$ as if you let $n=1$ and $m$ be infinitely large then its $1-\text{something infinitely small}$, but I’m not sure how to prove this?

How can I know the time difference between two cities almost at the same latitude?

Well I know that’s the earth rotation speed is: $v=1669.756481\frac{km}{h}$ I have two cities New York, Madrid almost at the same latitude and the distance between them is: $d=5774.39$ $km$ I know that’s : $\Delta t=\frac{d}{v}=3.4582$ $hours$ –> +3:27 But after I calculated it I found it wrong (according to and ( time difference […]

Let $(a_n)_n$ be a convergent sequence of integers , what can we say about $(a_n)_n$

Let $(a_n)_n$ be a convergent sequence of integers , what can we say about $(a_n)_n$? (I don’t understand what is meant by this question)