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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1answer
88 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 ...
0
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3answers
42 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_{...
0
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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 ...
1
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1answer
17 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 ...
2
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2answers
66 views

If ‎$‎f‎$ ‎is ‎a‎ convex function on ‎$‎[a, b]‎$‎, then ‎$‎f‎$ ‎is ‎increasing ‎on ‎$‎[a, b]‎$‎.‎

Let ‎$‎f:[a, b]‎‎\rightarrow ‎‎\mathbb{R}‎‎‎‎$ ‎be a ‎function. My ‎q‎uestion is:‎ If ‎$‎‎‎f‎$‎‎‎‎‎‎‎‎‎ ‎is a ‎convex function‎‎‎‎ ‎and ‎‎$‎f(x)\geq f(a)‎$ ‎for ‎all ‎‎$‎a\leq x\leq b‎$‎, then ‎$‎f‎$ ...
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2answers
76 views

A question from real analysis?

Let $A$ and $B$ be two non empty sets of real numbers such that $A \cup B = (0,1)$ then this implies that $ \inf(A)\inf(B) = 0 $ and if the claim is not true then give an example to support your ...
0
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0answers
44 views

Equivalent definitions of convergence in $\mathbb{R}$ and convergence in $\overline{\mathbb{R}}$

The usual metric space definition of convergence in $\mathbb{R}$ as a metric space is (1) $\lim \limits_{n \to \infty} x_{n}=x$ iff $\forall \epsilon>0$ $\exists N_{\epsilon}$ s.t. $n \geq N_{\...
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0answers
18 views

Lyapunov analysis on dynamical systems defined on a sphere

Related to the existing (unanswered) question (Discrete Dynamical Systems on Manifolds), but I'm interested in a bit more specific situation. Suppose we have a smooth dynamical system on a sphere $\...
2
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1answer
60 views

An integral inequality about $f(f(x))$

Let $f:[0,1]\rightarrow \mathbb{R}$ be continuous and monotonically increasing. Assume $f(0)=0$,$f(1)=1$. Prove:$$\int_0^1{f\left( f\left( x \right) \right) dx}\le 2\int_0^1{f\left( x \right)}dx $$...
1
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2answers
40 views

Convergent subsequence of $\sin(n^2)$

What can we say about the convergent subsequences of $\sin(n^2)$ whose existence is guaranteed by the Bolzano-Weierstrass theorem? Can we, as a corollary, claim that for all $\epsilon \gt 0$ there ...
1
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3answers
47 views

Define $f : L2 \rightarrow \mathbb{R}$ by $f(x) = \sum_{n=1}^{\infty} \frac{x_n}{n}$. Is $f$ continuous?

Problem I came across studying for qualify exams. I think I have it, if $x^k$ is a sequences of sequences in L2 and it converges to $x$ in L2, then $d(x^k_n, x_n) \rightarrow 0$ as $k \rightarrow \...
4
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1answer
37 views

Existence of nonsimilar matrix with same properties.

I wanted to know Is there exist 2 nonsimilar matrices with all algebraic properties same? I think there exists such pair as otherwise there we necessary sufficient condition of diagonalisibily using ...
4
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3answers
102 views

Definition of Open set.

I'm confused about the definition of Open set "A set Z is open, if for every element z in Z there's a r>0 such that there's an open ball contained with radius r in Z" I read the proof about "An open ...
0
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2answers
38 views

$\varepsilon$ Property of Supremum Doesn’t Hold for All Subsets of $\mathbb{R}$?

Let the set $S$ be $S = [0,2] \cup \{5\}$, the union of $[0,2]$ and $\{5\}$. This is a subset of $\mathbb{R}$? It has a supremum, which is 5. Although $4 < 5$, there is no member $x \in S$ such ...
0
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2answers
57 views

Prove the series $\sum_{n=1}^\infty (n(f(\frac{1}{n}) - f(-\frac{1}{n})) - 2f'(0)) $ converges

Prove the series $\sum_{n=1}^\infty (n(f(\frac{1}{n}) - f(-\frac{1}{n})) - 2f'(0)) $ converges where $f$ is defined on $[-1,1]$ and $f''(x)$ is continuous. I already have a solution for this but I ...
0
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0answers
34 views

Integration of periodic function

Let $a$ a T-periodic function. It is obvious that we have $$n\int\limits_0^T a (s) = \sum\limits_{i = 0}^{n - 1} {\int\limits_{iT}^{(i + 1)T} a (s)ds} $$ I want to estimate the following quantity$$\...
0
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0answers
34 views

Equivalence of measurability? What is the advantage of each of these?

So i am reading Royden and he introduces measurability via Cartheodory. $E \subset \mathbb{R}$ is measurable $\iff$ for any set $A \subset \mathbb{R}$ $m^*(A) = m^*(E\cap A) + m^*(E\cap A^c)$. ...
0
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1answer
21 views

Counterexample to Uniform Convergence of Convex Functions

It is stated here that if a sequence $f_n : [a,b] \to \mathbb{R}$ of convex continuous functions converges pointwise to a continuous function $f$, then the convergence is in fact uniform. The ...
0
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2answers
71 views

Is there a function f with $ f (x) $ divergent for large x and slower than logarithmic growth rates?

Problem: Is there a function $f:[0,+\infty]\to \mathbb{R}$ with the following conditions: $f(x)\ge 0$ for $x>C$ where $C$ is constant. $\lim_{x \to \infty} f(x)=+\infty$ $\lim_{x\to +\infty} \...
0
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1answer
33 views

Prove: If $D<2^{o(\sqrt{\log n})}$, then $D^{\lceil{\sqrt{\log n}} \rceil} \leq n$

A proof I am reading makes the following claim: Let $D,n$ be positive integer parameters. If $D<2^{o(\sqrt{\log n})}$, then $D^{\lceil{\sqrt{\log n}} \rceil} \leq n$. Why should this be?? My ...
7
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2answers
148 views

Interesting relation derived from the identity $\sin^2 x + \cos^2 x = 1$

Every one knows that $$\sin^2 x + \cos^2 x = 1.$$ It is also well known that $$\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n + 1}}{(2n+1)!},\quad\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(...
-1
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0answers
43 views

Every rational solution fo a polynomial with integer coefficients are integer solutions [duplicate]

I am trying to prove that for a generic polynomial in the form of xn + an-1xn-1 + ... + a1x + ao = 0 that any rational solution with integer coefficients ak must be an integer solution. I believe ...
0
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3answers
83 views

What does it mean to integrate a complex function over a real domain?

In complex form, we know that the $n$-th Fourier coefficient of a function $f$ is given by $$\int_{-\pi}^{\pi} f(\theta)e^{-in\theta} d\theta.$$ My question: What exactly does it mean to integrate ...
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0answers
62 views

Textbook Error? $\big [e^{\int_{0}^{x} e^{-t^2} dt} \big ]' = \int_{0}^{x} e^{-t^2} dt \cdot e^{-x^2} $ [on hold]

Textbook's Answer $ \big [e^{\int_{0}^{x} e^{-t^2} dt} \big ]' = \int_{0}^{x} e^{-t^2} dt \cdot e^{-x^2} $ My Answer $ \big [e^{\int_{0}^{x} e^{-t^2} dt} \big ]' = e^{\int_{0}^{x} e^{-t^2} dt} \...
1
vote
1answer
34 views

$L^p$-norm of the remainder of $L^2$-bounded sequence

We have a sequence of functions $(u_n)$ bounded in $L^2(M)$, where $M$ is a compact set in $\mathbb{R}^n$. We represent the sequence $(u_n)$ in the orthonormal basis $\{e_k\}_{k\in N}$ of $L^2(M)$: $$ ...
1
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0answers
46 views

Sufficient condition for convergence of the sequence $(1/a_n)$

I came across a question about convergence of sequences which I found interesting, but I would like comments on whether I have answered it correctly. I shall reproduce the question below. Let $(...
2
votes
3answers
88 views

The only subspace of $\mathbb{R}$ that is homeomorphic to $\mathbb{R}$ and complete (with the restricted metric) is $\mathbb{R}$

The only subspace of $\mathbb{R}$ that is homeomorphic to $\mathbb{R}$ and complete (with the restricted metric) is $\mathbb{R}$. My work- Let $A$ be a subspace of $\mathbb{R}$ such that $A$ is ...
5
votes
1answer
48 views

Conditional and absolute convergence of an integral

If you know that $$\int\limits_x^{+\infty} e^{-t^2}\mathrm dt=e^{-x^2}\biggl(\frac{1}{2x}+o\biggl(\frac{1}{x^2}\biggl)\biggl)$$ examine conditional and absolute convergence of this integral: $$\int\...
3
votes
1answer
52 views

Can we approximate elements in $L^2$ via smooth maps while preserving a pointwise constraint on the derivatives?

Let $\mathbb{D}^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball. Suppose we are given an open connected* subset $U \subseteq \mathbb{D}^n$ of full measure in $\mathbb{D}^n$, and a ...
-1
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1answer
40 views

Prove continuity for an integral function [on hold]

Let $f$ a continuous real function defined on $[0,1]$ be given. It's asked to prove continuity in $x \in [0,1]$ for the function $$x \to \int_{0}^x \frac{f(t)}{(x-t)^{\frac{1}{2}}}dt$$ Thanks in ...
4
votes
3answers
140 views

Under what conditions do we have $\int_{0}^{\infty} |f(x)|^2 dx \leq C \int_{0}^{\infty} x^2 |f^{\prime}(x)|^2 dx$?

I have been trying to prove the inequality $$\int_{0}^{\infty} |f(x)|^2 dx \leq C \int_{0}^{\infty} x^2 |f^{\prime}(x)|^2 dx$$ for some constant $C$, under the most general set of assumptions ...
1
vote
1answer
29 views

Necessary and sufficient condition for convergence almost surely by Borel-Cantelli lemma

Given a sequence of random variables $\{X_n \}_{n=1}^\infty$, the first Borel-Cantelli lemma tells us that if there is a positive sequence $\{ a_m \}_{m=1}^\infty$ for which: $$ a_m \overset{m\...
8
votes
4answers
128 views

If $L^1$ average of $f$ is smaller than $t^2$ then $f=0$ a.e.

Suppose $f\in L^1(\mathbb R)$ and there exits $\delta>0$ such that $$\|f(\cdot + t)-f\|_{L^1}\leq |t|^2$$ for all $|t|\leq \delta$, where $f(\cdot + t)$ is the function $x\mapsto f(x+t)$. Show ...
1
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4answers
114 views

Prove that if $f=f'$ then $f$ is monotone.

Suppose I do not know that $e^x$ solves the equation $$f'(x)=f(x),\;\;\;x\in\mathbb{R}.$$ I am just given this equation and want to see if $f$ is increasing. Is there a way to prove that $f$ is ...
0
votes
2answers
51 views

The Definition of Derivative of function in Analysis on Manifold by Munkres

so i've been reading Analysis on Manifold by Munkres and in page 43, there is a definition of derivative of $f :\mathbb{R}^m \to \mathbb{R}^n$ which is: "Let $A\subset \mathbb{R}^m$, let $f:A\to \...
1
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0answers
30 views

Uniformly bounded sequence of non-decreasing functions has a convergent subsequence

I came across the following problem in an old notebook. Let $f_n:\mathbb{R}\to [0,1]$ be a sequence of non-decreasing functions, that is, $f_n(x)\le f_n(y$ whenever $x\le y$ for all $n$. Show that ...
0
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2answers
62 views

If $~\lim_{x\to a}(f(x))^4 = 0~$, prove that $~\lim_{x\to a} \bigl(−2f(x)\bigr) = 0~$.

Let $~a ∈ \mathbb R~$ and let $~f: \mathbb R → \mathbb R~$. If $~\lim_{x\to a}(f(x))^4 = 0~$ then provide a complete $~δ − ε~$ proof that $~\lim_{x\to a} \bigl(−2f(x)\bigr) = 0~$. Can somebody help ...
0
votes
2answers
54 views

Differentiable function except at integers.

$f:\mathbb{R}\to \mathbb{R}$ is a continuous function which is not differentiable only at the integer points, then a) f is bounded b) f is unbounded c) f is uniformly continuous d) f may not be ...
0
votes
1answer
18 views

Bound on positive linear functional on $C_c(X,C)$ over a compact set $K$.

Let $X$ be a locally compact hausdorff space. Then $X$ is normal. Let $K$ be a compact subset of $X$. Denote $C_c(X)$ the set of continuous functions valued in complex number with compact support. Now ...
1
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0answers
26 views

Geometry of Envelope form definition

I had read about the envelope of the family of the curve. It is defined as a curve which is tangent to each member of the family at a single point and it is union of all such points. To find envelope ...
2
votes
4answers
73 views

Exercise 2.5.6 of Understanding Analysis

Let $(a_n)$ be a bounded sequence and define the set S = $\{ x \in R : x < a_n$ for infinitely many terms of $a_n\}$. Show that there exists a subsequence $(a_{n_j})$ converging to $s$ = sup $S$. ...
1
vote
2answers
96 views

How to compute $\sum_{k=1}^{\infty}{(\zeta(2k)-1)}$

How to compute $\sum_{k=1}^{\infty}{(\zeta(2k)-1)}$, where $\zeta(s) :=\sum_{k=1}^{\infty} \frac{1}{n^s}$ with $s>1$. Here's my process, what am I doing wrong?:
0
votes
2answers
60 views

If $x_n=0$ infinitely often, does $\lim_{n\to \infty }\frac{y_n}{x_n}$ makes sense?

I have a sequence $(x_n)$ s.t. $x_n=\frac{1}{n}$ if $n$ even and $0$ if $n$ odd. We set $y_n=\sin(x_n)$. I have to compute $$\lim_{n\to \infty }\frac{y_n}{x_n}.$$ It's an exam question... intuitively,...
1
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2answers
80 views

The series $\sum_{n=1}^{\infty} \frac {\cos n}{2n^{\alpha}}$ converges for $\alpha \in (0,1)$.

I'm trying to solve this problem: Let $\alpha>0$ and $a_{n}=\frac{\cos n}{2n^{\alpha}}$ for all $n\in\mathbb{N}$. Prove that the series $\sum_{n=1}^{\infty}a_{n}$ coneverges. For $\alpha>1$, we ...
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0answers
53 views

Find a function which satisfy given inequality [closed]

Let $g:\mathbb{R}^{+}\rightarrow\mathbb{R}$ satisfies $$ |g'(x)|\leq \frac{C_1}{1+x} $$ for some constant $C_1$ and $$ |g''(x)|\leq \frac{C_2}{1+x^2}. $$ for some constant $C_2$. Find a function $f:\...
2
votes
2answers
57 views

Does $(f_n)=(n\sin(\frac{x}{n})-x)$ converge uniformly on $[-a,a]$ for $a\geq0$?

I'm trying to solve the next problem: Let $\left(f_{n}\right)_{n\in\mathbb{N}}$ be a sequence of functions such that $f_{n}\colon\mathbb{R}\to\mathbb{R}$ is given by $f_{n}\left(x\right)=n\sin\left(\...
1
vote
3answers
38 views

Construct $b_n$ so that $\sum_{n=0}^{\infty}b_n=B$

There are given two real numbers, convergent series $\sum_{n=0}^{\infty}a_n=A$ and $\sum_{n=0}^{\infty}c_n=C$, such that $a_n<c_n\:\forall n\in\mathbb{N}$. Let $B\in(A,C)$; Construct $b_n$ such ...
1
vote
0answers
23 views

Mean field games and approximate Nash equilibria

I am learning mean field games (MFG) through the notes by Cardaliaguet: https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf. I have a question about a step in theorem 3.8 on page 17. Let ...
0
votes
1answer
25 views

Topology of point wise convergence

I encountered this topology in the exercise below. I was wondering if there is any connection to algebraic geometry? This looks like algebraic geometry but instead of points we are dealing with ...