# Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with 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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### What's the supremum of $\{x \in F: -2 < x \le 2\}$?

The answer in my book defines one of the variables in terms of $\max$ function which seems to cause some sort of weird inconsistency. Instead I rewrote the proof as that below. I am asking if that ...
0answers
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### Proof Chebyshev inequalities

I have to work my way through the chebyshev inequalities and can't find any references. Can someone recommend books to me or send me the link to proof? Thank you in advance. Best regards Necip I have ...
2answers
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### Integral Table Result

To prove the result $$\int_{1}^{e}\frac{\ln(x)}{(\alpha +\ln(x))^{(\alpha+1)}}\,dx = \frac{e}{(\alpha+1)^\alpha} - \frac{1}{\alpha^\alpha}$$ I have used the substitution $t = \alpha +\ln(x)$ which ...
1answer
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### Prove that $\sum_{i=0}^{m_n} \Delta x_i^n\varepsilon _{i,n}(\Delta x_{i}^n)\to 0$ when $n\to \infty$.

Introduction This question is inspired from one of my other question here, and at the end, I'm really not convinced by the answer that I accepted. However, I recognize that the notations are a bit ...
2answers
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### Study the convergence of the series [on hold]

I need to study the convergence of the following series: $$\sum_{n=1}^{\infty}(\frac{1\cdot3\cdot5\cdot\cdots\cdot(2n+1)}{1\cdot4\cdot7\cdot\cdots\cdot(3n+1)}x^n)^2$$ I tried to expand that term and ...
0answers
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### Unique time reachability for two symmetric ODE

We consider the following ODE $y'(t)=f(t,y(t))$ with initial conditon $y(0)=0$ and $y'(t)=-f(t,y(t))$ with initial condtion $y(0)=1$ and denote $y^+$ and $y^-$ respictevely. Assume that f is ...
1answer
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1answer
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### Let $\epsilon$>0. Let $f$ from $[0, +\infty)$ to $[0, +\infty)$ is differentiable and $f'>0$. [on hold]

Why is $$\lim_{x\rightarrow\infty}\frac{f^{1+\epsilon}(x)}{x*f'(x)}=\infty?$$ The L'Hopital rule is not useful here.
2answers
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### Comparing $L^p$-norms of a function for two different values of $p$.

Let $(S,\Sigma,\mu)$ be a measure space with $\mu(S)<\infty$. Let $p\in(1,\infty]$ and suppose $f\in L^p(\mu)$. A simple application of the Holder's inequality shows that for any $0<q<p$, $f$ ...
0answers
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1answer
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### Volume of a union of triangles

Let $t \in \mathbb{R}$ and consider the points $$A(t) = (t,t^3,t), \, B(t) = (t,t,t), \, C(t) = (0,2t,t)$$ Find the volume of $$\Omega = \bigcup_{t \in [0,1]} T(t)$$ where $T(t)$ is the triangle ...
3answers
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### Doubt regarding definition of converging sequence.

I am currently studying about converging sequence in my Real Analysis class. The definition of a converging sequence is A sequence of real numbers converges to a real number a if, for every ...
2answers
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### Proof Verification - Archimedean Property

I am self-learning real analysis and learning to write proofs. I am trying to prove the Archimedean Property and would like to check if my attempt at a proof is correct and how to improve my proof ...
2answers
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### Non standard solution to $f(x) = \frac{1}{2}\Big(f(\frac{x}{2}) + f(\frac{1+x}{2})\Big)$

This functional equation appears in the following context. Let $\alpha\in[0,1]$ be an irrational number (called seed) and consider the sequence $x_n=\{2^n \alpha\}$. Here the brackets represent the ...
2answers
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### Limit at infinity of the differential implies uniform continuity

Assume that $f: \mathbb{R}^{+} \to \mathbb{R}$ is a differentiable function, with $\lim_{x\to\infty}f'(x)=1$. Is $f$ then uniformly continuous on $\mathbb{R}^+$? I'm not really sure where to get ...
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### rudin books problem clarification [on hold]

"If a ordered set has least upper bound property then its every subset of ordered set have a same upper bound or not ?" doubt in rudin "mathematical analysis" page number--5
3answers
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### Does there exist a sequence that has countably infinite convergent subsequences?

I know that for every natural number $n$ there is a sequence with exactly $n$ convergent subsequences, where I consider two subsequences to be the same if they are equal as sequences (even if they ...
1answer
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### Confirmation about one simple inequality [duplicate]

Suppose A & B are sets of reals.Then is it true that inf{A+B}$\leq$inf{A}+inf{B}.I consider all the cases like A contains all positive and B contains all positive,then alternate positive and ...
2answers
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### lower bound of the difference between two numbers

There are two real numbers $x_1$ and $x_2$, and $x_1$ is bounded $[ay, by]$ and $x_2$ is bounded by $[cy, dy]$ where $a$, $b$, $c$, and $d$ are all positive numbers. Is there a lower bound for the ...
2answers
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### Is set of matrices form $M_{3\times 3}(\mathbb R)$ with rank 1 is closed or open?

Is set of matrices form $M_{3\times 3}(\mathbb R)$ with rank 1 is closed or open ? I know that $M_{3\times 3}(\mathbb R)$ can be viewed as $\mathbb R^9$ with the euclidean norm on it. But I do not ...
3answers
48 views