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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9 views

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 I accepted. However, I recognize that the notations are a bit confuse ...
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2answers
19 views

Study the convergence of the series

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 ...
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0answers
12 views

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 ...
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18 views

Does $\lim_{x\to 1^-}\sum_{n=0}^\infty a_n x^n\neq\infty$ implies $-\infty<\liminf\sum_{n=0}^N a_n\le\limsup\sum_{n=0}^N a_n<+\infty$?

Notation. In the text below, I consider series of real terms and use the following terminology. If $\lim_{N\to\infty}\sum_0^N a_n=\lim_{N\to\infty} s_N=\sum_0^\infty a_n=\pm\infty$ the series is ...
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1answer
48 views

Can a function from $(0,1)$ to $(0,1]$ be one-to-one and onto? [duplicate]

Does there exist a function from $(0,1)$ to $(0,1]$ both one-to-one and onto, not necessarily continuous? I couldn't think of any. Any help would be appreciated! Thanks,
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2answers
34 views

Does such a differentiable function exist?

Does there exists a differentiable function $f: \mathbb{R} \to \mathbb{R}$ with $f'(0)=0$ and the existence of a sequence $(x_n)_n$ in $\mathbb{R}$ such that $x_n \to 0$ implies $f(x_n)\to \infty$. ...
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0answers
8 views

Why does prior predictive distribution make sense?

I am trying to understand why prior predictive distribution makes sense. Let $X,\Theta,Z$ be random variables on a probability space with the probability measure $P$, and assume that $\Theta$ is ...
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1answer
26 views

Is this Change of Variable Theorem fine as it is, or should we specify the domain that $f$ is continuous on?

One of my instructors provided me with the following Change of Variable Theorem: Let $a < b$. Let $f$ be a continuous function. Let $g$ be a function with a continuous derivative on $...
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1answer
44 views

Let $\epsilon$>0. Let $f$ from $[0, +\infty)$ to $[0, +\infty)$ is differentiable and $f'>0$.

Why is $$\lim_{x\rightarrow\infty}\frac{f^{1+\epsilon}(x)}{x*f'(x)}=\infty?$$ The L'Hopital rule is not useful here.
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27 views

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 $0<q<p$, $f$ ...
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38 views

Ordinary differential equation solution ideas.

Let $f$ be a differentiable function and $\mu>0$. Is there an explicit solution $y$ of the following first order ODE? $$ |y'(x)| = \exp\left(-\mu\sqrt{f^2(x) + (y(x)-x)^2}\right), \ \ x\in \ ]-\...
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50 views

If the cardinality of $f^{-1}$ is at most $f(x)^2$ then $f$ is differentiable almost everywhere.

I came across the following problem in a prelim question paper. The question as stated seems meaningless to me, I am adding the picture so as to avoid any error from my end. My case with the above ...
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2answers
19 views

Counterexample of a sequence in $l^2(N)$ which is pointwise convergent but not 2-norm convergent

Here $l^2(N)$ is the set of real sequences $x_{n}:N\rightarrow R$ with $\sum{|x_{n}|^2}<+\infty$. I need a sequence in this space that convergences pointwise but not in the norm $\|x_{n}\|=(\sum {|...
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1answer
24 views

Does the notion of a contraction operator depend on the norm which induces the metric?

Let $\lVert \cdot \rVert_a$ and $\lVert \cdot \rVert_b$ denote the $\ell_a$ and $\ell_b$ norms on $\mathbb{R}^n$ ($a,b$ are positive integers or $\infty$, $a \neq b$). Let $M_n(\mathbb{R})$ be the ...
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2answers
28 views

Is following function $\exp(x) + \exp(y)$ super-coercive?

Consider the function $f(x,y) = \exp(x) + \exp(y)$, $(x, y) \in \mathbb{R}^2$. I want to know if it is true that,$\exp(x) + \exp(y)$ super-coercive, that is, $(\exp(x) + \exp(y))/\|(x,y)\|_2 \to \...
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4answers
49 views

A function f is continuous on $\mathbb{R}$ and $f(n)=0$ for all integers $n$, then $f$ may not always uniformly continuous.

I tried to find such example that f is not uniformly continuous. But if I think it by graphically, then this function return to x axis every time at integers with a continuous arc. I don't understand ...
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0answers
15 views

measureability of a function and its relation to the completion of the measure

I am trying to understand the need for proposition 2.12 in Folland. Proposition 2.11: The following implications are valid iff the measure is complete: a. If $f$ is measurable and $f=g$ $\mu$-a.e., ...
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1answer
29 views

Showing a linear map is continuous

This is exercise 6, part b) in chapter 5 out of Folland's Real Analysis textbook. Let $X$ be a finite dimensional vector space (over $\mathbb{R}$) with basis $\{e_i\}_{i = 1}^{n}$. Equip $X$ with the ...
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3answers
36 views

Unit sphere difficult isometry

Let $S^2$ be the unit sphere $x^2+y^2+z^2=1$ and let $f:\mathbb{R^2} \to S^2$ , $(u,v) \to (x,y,z)$. The point $(x,y,z)$ is the second point where the line that passes through $(u,v,0)$ and $(0,0,1)$ ...
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21 views

Proof of non-strictly convexity of $L^1$ and $L^\infty$ [on hold]

We define the following norms: $$\|f\|_{L^p} := \left(\displaystyle\int |f|^p\,d\mu\right)^{\frac{1}{p}},\quad p\geq 1~,$$ and $$\Vert f\Vert_{L^\infty}:=\operatorname{esssup}|f|~.$$ ANY AGAINST ...
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2answers
24 views

Extending a $C^{1}([0, 1/2])$ function to be $C^{1}([0, 1])$ while preserving upper and lower bounds

Suppose $f \in C^{1}([0, 1/2])$ such that $1 \leq f(x) \leq 2$ for all $0 \leq x \leq 1/2$ and $f'(0) = 0$. Does there exist a $g \in C^{1}([0, 1])$ such that $1 \leq g(x) \leq 2$ for all $0 \leq x \...
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0answers
84 views

Proof check: Baby Rudin 7.25

I am mostly concerned with part (e) since my version disagrees with the key that I checked my work against (I think the key's solution to this part is wrong). As I understand it, the functions $f_n$ ...
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1answer
18 views

How close is a countable generator related to separability?

Let $(\Omega, \tau)$ be a topological space. If the space is separable then exists countable $S\subseteq \Omega$ so that for any $V \in \tau$ and $v \in V$ there exists a $W \in \tau$ and $s \in S$ ...
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2answers
33 views

Why is it clear that $\sigma(\Omega_{1})\times \sigma(\Omega_{2})\subseteq \sigma( \Omega)$

Let $(\Omega_{1}, \tau_{1})$ and $(\Omega_{2}, \tau_{2})$ be topological spaces and $\tau:=\tau_{1}\otimes \tau_{2}$ while $\Omega:=\Omega_{1}\otimes \Omega_{1}$ I am attempting to brush up on my ...
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1answer
46 views

How to prove that a polynomial has one exactly one root using Banach's fixed point theorem?

So this question had two main parts that I got stuck on: Suppose that (X,d) is a complete metric space and $f : X \rightarrow X$ is a map. Parts a) & b) just asked for the definition of a ...
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11 views

Integral of asymptotic equal functions

I have lemma to prof about asymptotic equal functions, could you please advise me on it? Let $h_{i}: R^{2} \to R$ for $i=1,2$ such that $h_{1} (x,y) \underset{x \to a} {\asymp} h_{2} (x,y)$ for all $...
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1answer
28 views

Orthogonality condition for two complex-valued vectors $a,b \in \mathbb{C}^n$, iff $\operatorname{Re}\left\{a^* b \right\}=0$?

I am sorry for asking a basic question. Orthogonality condition for the two complex-valued vectors $a,b \in \mathbb{C}^n$ is iff $\operatorname{Re}\left\{a^* b \right\} = 0$? I thought the ...
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3answers
92 views

What is the meaning of $2^\sqrt{3}$? [duplicate]

What is the meaning of $2^\sqrt{3}$ ? one can explain $2^3=2\times2\times2$ But how can I explain $2^\sqrt{3}$ etc. ? How can I explain non-integer powers? I do not want the value. I need to know ...
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1answer
33 views

What is the gradient of $f(x) := \langle v , F(x) v \rangle$?

Let $F: \mathbb{R}^n \to S^n$ be differentiable function at point $x \in \mathbb{R}^n$. Where $S^n$ is the space of all symmetric matrices with usual Euclidian (trace) norm. Assume $v \in \mathbb{R}^n$...
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1answer
25 views

Infinite product proof without using gamma function [duplicate]

How do you show that the infinite product of $(3n+1)/(3n+2)$ converges to zero without using the gamma function (as I can tell the gamma function solution has already been presented here)? I've tried ...
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3answers
49 views

Prove $\exp{\frac{a+b}{2}}>\frac{1}{2}(\exp a + \exp b), a,b\in \mathbb{R},a \ne b$

Self-studying. Tools I considered and used: the rules for $\exp$, $\log$, the fact that $a>b\iff \log a > \log b$, i.e., $\exp$ is an increasing function, derivatives of same, Taylor expansions ...
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0answers
34 views

Doubt on some integral equality concerning $f'$ and the quotient $\frac{f(x)-f(x-\alpha)}{\alpha}$

I am stuck proving this equality $$ \int_{\mathbb{R}} \frac{f' (x-\alpha)}{|\alpha|^{2-\varepsilon}}\ d\alpha = C (\varepsilon) \int_{\mathbb{R}} \frac{f(x)-f(x-\alpha)}{\alpha} \frac{d\alpha}{|\alpha|...
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1answer
48 views

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 ...
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3answers
36 views

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 ...
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2answers
46 views

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 ...
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1answer
157 views

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 ...
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2answers
26 views

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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28 views

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
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4answers
71 views

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 ...
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1answer
31 views

Confirmation about one simple inequality

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 ...
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2answers
34 views

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 ...
3
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2answers
41 views

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 ...
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3answers
47 views

Find an equivalent sequence

Consider $ u_n = (n+1)^{1/n+1} - n^{1/n} $ Find an equivalent sequence at infinity. (meaning $ u_n / y_n \rightarrow 1 ) $ I tried doing : $ u_n = e^{ \frac{ln(n+1)}{n+1}}(1 - e^{\frac{ln(n)}{n} -...
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0answers
33 views

Determining the convergence/divergence of recursive sequence

Is it always possible to define a sequence using both $n^{th}$ term formula and recursion formula? For example: $a_1=3$, $a_{n+1}=a_n+3$ defines the sequence $\{3,6,9,12,...\}=\{3n\}$ I am asking ...
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2answers
49 views

Find out if $f(x,y)=\frac{1}{(1-xy)^2}$ is integrable over $[0,1]^2$

Find out if $f(x,y)=\frac{1}{(1-xy)^2}$ is integrable over $[0,1]^2$. If I could use Fubini: $$\int_0^1\int_0^1 \frac{1}{(1-xy)^2}dxdy=\int_0^1\frac{1}{1-y}dy=\infty $$\ But can't use Fubini, ...
3
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1answer
82 views

What is the realcompactification of the real line?

I've studied the definition of the Stone-Cech compactification by Munkres Topology. I have realized that we can't write the Stone-Cech compactification of the real line explicitly, we are just able to ...
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3answers
41 views

Orthogonal polynomials with respect to the weighting function $\omega(x)=\frac{x}{e^x-1}$

I'd like to build a family of orthogonal polynomials with respect to the weighting function $\omega(x)=\frac{x}{e^x-1}$ on $[0,+\infty)$, i.e the inner product $$<P_n|P_m>=\displaystyle\int_{...
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0answers
31 views

estimate $\mathcal{O}(\sqrt{n!})$ and $\mathcal{O}(\log{n!})$

How to estimate $\mathcal{O}(\sqrt{n!})$ and $\mathcal{O}(\log{n!})$. So for example $\sqrt{n!} = \sqrt{1}* \sqrt{2}* ....*\sqrt{n} \leq \sqrt{n}^n$ and $\sqrt{n!} \geq \sqrt{1}^n$. I can't find ...
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1answer
15 views

necessity of semi finiteness

As stated in this question sigma finite measures are semi finite. $\sigma$-finite measure and semi-finite measure I am interested in the question if we can weaken the sigma finiteness condition to ...
2
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1answer
40 views

Prove limit of the function $g = f-xf'$

Let $f:[0,\infty) \to \mathbb{R}$ be a $C^2$ function and $g :[0,\infty) \to \mathbb{R}$ be defined as $g(x) = f(x) -xf'(x)$. Prove that $f$ is convex if and only if $g$ is non increasing (and this is ...