Tagged Questions

Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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Partial sum of bounded series is Cauchy in $C(X, \mathbb{R})$?

I am reading a proof of Tietze's Extension Theorem and there was a claim that, given a sequence of functions $h_n(x) : X \to \mathbb{R}$ If $$G = \sum\limits_{n = 1}^\infty h_n(x)$$ Is bounded, then ...
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Continuous functions with orbit of period $3$

I would like to build some continuous functions $f : E \to \Bbb R$ (where $E \subset \Bbb R$ is an interval), such that $$\exists x \in E,\;\; [f(x)≠x≠f(f(x)),\;\; f^3(x):= f(f(f(x)))=x]$$ I tried ...
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Evaluating $\lim_{n\rightarrow\infty} n \int_{0}^{1} \frac{{x}^{n-2}}{{x}^{2n}+x^n+1} \mbox {d}x$

Evaluating $$L = \lim_{n\rightarrow\infty} n \int_{0}^{1} \frac{{x}^{n-2}}{{x}^{2n}+x^n+1} \mbox {d}x$$
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Showing $f(a) \in V$ but $f(x_n) \notin V$ for every $n$

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces and let $f: X \rightarrow Y$ be a function. Let $a \in X$ and suppose $f$ is not continuous in $a$. Prove that there exists an open subset $V$ in $Y$ ...
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How can I show this sequence $u_n$ is divergent: $u_n=\exp( n\log n-(n+\epsilon)\log(n+\epsilon))$

How can I show this sequence $u_n$ is divergent: $$u_n=\exp( n\log n-(n+\epsilon)\log(n+\epsilon))\quad n\in \mathbb{N}^*;\quad \epsilon \in (0,1)$$ My attempts: \begin{align*} u_n&=\exp( n\log ...
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How to Find a rational number between two irratonal number? [on hold]

Find the rational number between $\sqrt 2$ and $\sqrt3$. I try to solve by using some methods in my book but can not understand steeps.
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Is it possible to use Lusin's Theorem to derive Frechet's Theorem?

Frechet's theorem states that every measurable function $f$ on $\mathbb{R}$ is the limit of a sequence of continuous functions converging almost everywhere. Frechet's theorem is then used to prove ...
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Creating a “nice” R^2->R function from partial subdomains

This question is probably poorly defined, mainly because I am still not entirely sure what I need, and I just need some ideas to start with. I have a R^2 domain, with a set of given curves ...
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How smooth can non-nice associative operations on the reals be?

Suppose ${*}:\mathbb R\times\mathbb R\to\mathbb R$ is $\mathcal C^k$ and associative. Does it necessarily satisfy the identity $a * b * c * d = a * c * b * d$? For $k=0$ the answer is "no" -- a ...
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Density of $C^\infty$ functions on $C[-1,1]$

I'm reading Peter Lax functional analysis, and the author assess that that there exits for every $f \in C[-1,1]$ so that $f(0)= 0$ a $C^\infty$ function $g$ such that the distance between $f$ and $g$ ...
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Prove $s_1 = 4, s_{n+1} = \sqrt{3s_n -2}$ Convergent Sequence

Prove that this sequence is convergent, as $n -> \infty$ $s_1 = 4, s_{n+1} = \sqrt{3s_n -2}$ In our lecture, our teacher explained proving the above using the following theorem: A ...
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On Riemann's series of functions

The questions given below are devoted to study properties of some series of functions which are directly connected with Riemann's Conjecture(see, for example Again near at Riemann ...
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Prove a composition of two functions is meaurable [duplicate]

Let $f,g:[0,1]\rightarrow [0,1]$ be measurable functions.Is $g\circ f$ measurable or not? The composition is definitely measurable from the axiom definition of measurable function. But if we want to ...
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Prove: if $s_n$ is Cauchy Sequence, then $s_n$ converges as $n \to \infty$

How to prove the following: if $s_n$ is Cauchy Sequence, then $s_n$ converges as $n \to \infty$ Using the following theorem: A Cauchy Sequence is bounded. Assumption: 1. $s_n$ is ...
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A Question on Normal Operators

Let $T$ be a compact operator on a Hilbert space $H$. I want to prove the following: If there exists Orthonormal Basis from the eigen vectors of $T$, then $T$ is normal operator.
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Proving that $\int_a^bf^pd\alpha=0$ implies $\int_a^bfd\alpha=0$ for $f \in \mathcal{R}(\alpha)$ and $f\ge0$ on $[a,b]$

The question posed may seem trivial to many. But I couldn't find a trivial solution to above within the theory upto Riemann-Stieltjes integral. I woudn't be considering arguments from Lebesgue theory ...
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Taylor expansion of $\left(\int_{t}^{t+\Delta t}a(t')dt'\right)$

$$\left(\int_{t}^{t+\Delta t}a(t')dt'\right), a(t) \text{ is scalar}$$ Is the taylor expansion of $\left(\int_{t}^{t+\Delta t}a(t')dt'\right)$ at $t$ = \left(\int_{t}^{t+\Delta t}a(t')dt'\right) = ...
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How do I prove two definitions of the variation of a measure are equivalent?

Let $(X,\Sigma)$ be a measurable space and $\mu:\Sigma\rightarrow \mathbb{C}$ be a complex measure. Define $|\mu|(E)$ as the supremum of $\sum_{n=0}^\infty |\mu(E_n)|$ where $\{E_n\}$ is a mutually ...
Let $f : X \rightarrow \mathbb{R}$ be a continuous function and let $a \in \mathbb{R}$. Determine whether the following statements are true and false. Prove you answer. i) $\{x \in X :f(x) \leq a\}$ ...