# Questions tagged [real-analysis]

For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.

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### How to show that the limit of a sequence is not equal to some value?

In the second chapter of the book Understanding Analysis by Abbott, Example 2.2.6 proved that picking $N>\frac{1}{\varepsilon}$ suffices to prove the convergence of $\frac{n+1}{n}$ to the number $1$...
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### Completely monotone function is analytic

I want to prove the following. Let $I:=[a,b)$ be finite interval, $f$ is completely monotone on $I$. Then it can be extended analytically into the complex z-plane ($z=x+iy$), and the function $f(z)$ ...
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### How to give this sum a bound?

Let $x,y\in\mathbb{Z},$ consider the sum below $$\sum_{x,y\in\mathbb{Z}\\ x\neq y}\frac{1}{|x-y|^{4}}\Bigg|\frac{1}{|x|^{4}}-\frac{1}{|y|^{4}}\Bigg|,$$ is there anything I could do to give this sum a ...
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### Infinite Summation of Almost Sure Convergent RVs

Suppose we have random variables $X_{n,i}$ for $n\geq1,i=1,\dots,b_n$. Here we suppose $b_n$ is non-decreasing. We know that for each $i$, the sequence $X_{n,i}$ converges to $X_i$ almost sure. Now we ...
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### The radius of convergence using root test

Is the root test sufficient for finding the radius of convergence of any power series ,i.e, does it determine the whole area where a power series converges? if the answer is no can you give an example ...
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### What is this set? Unknown baire generic set

I'm interested in the set given in this question. For clarity, set $$S_{\epsilon} = \cup_{j = 1}^{\infty} \left( q_j + \frac{\epsilon}{2^j}, q_j - \frac{\epsilon}{2^j}\right)$$ where $\\{q_j\\}$ is ...
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### Missing argument in the proof of the Levy-Khintchine representation .

In one proof of the Levy Khintchine representation of the Laplace exponent of subordinators, the following argument is used: "Assume $f_n : [0,\infty) \to [0,\infty)$ is a sequence of non-...
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### How to get the following estimate of integral invoving Airy function

$$\mbox{Define}\quad G(x,y)=\frac{Ci\left(\gamma(x-y_c)\right)Ai\left(\gamma(y-y_c)\right)}{\epsilon\gamma},$$ where $y>x,$ $y_c$ is a complex number such that $\Re(y_c)>0,$ $\epsilon$ is a ...
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### Diophantine approximation and asymptotic for $\dfrac{1}{\sin(n\pi\sqrt{3})}$

I have this exercise and i proved the lemma but i couldn't use it to prove the asymptotic formula i couldn't plug the sin function into the inequality because it change variations maybe some choice of ...
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### fractional power function inequality

I am facing an issue to prove if its true the following inequality and any help is greatly appreciate. I used this inequality $x^\alpha \le \alpha x+1-\alpha$ but still I could not get the result. Let ...
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### Subset of index that minimizes a sum of real values

Given a series of real numbers $c_1, \ldots, c_n$ with $n \in \mathbb{N}$, is there an algorithm or method to find the subset of indices such that the absolute value of the sum of the values within ...
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### Converse of bounds on the spectrum of a Toeplitz matrix

The following is from Robert M. Gray's review (https://ee.stanford.edu/~gray/toeplitz.pdf): Lemma 4.1 Let $\tau_{n,k}$ be the eigenvalues of a Toeplitz matrix $T_n(f)$. If $T_n(f)$ is Hermitian, then ...
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### Smallest value of $k$ for which a function approaches $0$ as $x$ goes to $\infty$

I was playing around with factorials on desmos and trying to find some inequality between $x!$ and $\sqrt[x]{x}!$. After a bit I formulated the following question: What is the smallest value of $k$ ...
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### Regularity for computing the first variation

I am having a trouble understanding the regularity needed to compute the first variation for the Euler-Lagrange equation for the functional $$F(u) = \int f(u) dx$$ Suppose $u:U \to \mathbb{R}$ for ...
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### Taylor's Theorem for functions on the Rational Numbers

I have been looking for different proofs of the Taylor's Theorem with Peano form of the remainder. But all proofs I have found use some form of the Mean Value Theorem or L'Hôpital's Rule (which also ...
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### If $f$ and $\frac{\partial^2 f}{\partial x^2}$ are continuous, then $\frac{\partial f}{\partial x}$ is continuous?

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a function such that both $f$ and $\frac{\partial^2 f}{\partial x^2}$ are continuous on $\mathbb{R}^2$. Is then $\frac{\partial f}{\partial x}$ also ...
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### Why is a lot of Fourier analysis done on an annulus?

I am studying harmonic analysis from these lecture notes and a lot of results and definitions always assume that the Fourier transform of a function has support in an annulus or a ball. The same ...
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### Fractional regularity and the quotient $\frac{f(x)-f(y)}{|x-y|^\alpha}$

Holder, Sobolev, and Besov Spaces are often used to measure regularity, and in particular, fractional regularity. On one hand, the relation between their respective norms and regularity/...
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### Solution of the "reciprocal of the heat equation"?

I was playing around with the heat equation in one dimension and tried to guess what the solution to homogenous boundary conditions and a sine wave as initial condition on the interval $0<x<\pi$ ...
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### Can an unbounded strictly increasing sequence be convergent? [duplicate]

I was doing excercises on convergence of sequences of real analysis but I came up with a problem I don't know how to prove. Note: The book has not shown Cauchy sequences yet and I don't know what they ...
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### A problem on dirt displacement

Definition. Given a function $f\in L^1(\mathbb{R})$ such that $xf\in L^1(\mathbb{R})$, the quantity $\int_\mathbb{R}xf(x)\,dx$ is called the unnormalized center of mass of $f$ and is denoted $UCM(f)$. ...
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### Rudin's RCA 2.24 theorem : Lusin's theorem

As precised in the title of my question, the context is the book of Walter Rudin : Real and Complex analysis. And especially the proof of theorem 2.24 (Lusin's theorem) which I put below. I have a ...
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