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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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Finishing the proof of the triangle inequality of Hausdorff metric

currently I am trying to show that a certain type of Hausdorff metric satisfies the following triangle equality and I am stuck. Setup: Take $(X,d)$ as metric space. Denote by $C(X)$ the set of closed ...
a.s. graduate student's user avatar
0 votes
0 answers
13 views

If $G\in C([0,1])$ and strictly increasing, can we find a sequence $G_n\n C^{\infty}$ with uniformly equicontinuous density?

Let $G:[0,1]\rightarrow [0,1]$ be a strictly increasing and continuous cdf with $G(1)=1$. I have proven some property for $G\in C^{\infty}([0,1])$ that relies on the continuity of $g(x)=G'(x)$. I hope ...
djsteve's user avatar
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0 votes
1 answer
60 views

How to visualize triangle using a software

How to visualize (using some plotting software) the triangle inequality, $|x-z|\le|x-y|+|y-z|$, where $x,y$ and $z$ belong to some finite intervals. I am asking for general way, for example, $(x-z)^2\...
stephan's user avatar
  • 373
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0 answers
41 views

Prove every Cauchy sequence in R is bounded

Here's my attempt to prove every Cauchy sequence in $\mathbb{R}$ is bounded, I would like to see if there are any flaws in it. Proof: Let ${(x_n)}$ be a Cauchy sequence in $\mathbb{R}$. Now let's ...
Jackson Smith's user avatar
3 votes
2 answers
81 views

Set with elements of the form $𝑓_𝑛:[0,1] β†’ ℝ$ defined as $𝑓_𝑛 (𝑥) = 𝑥^ 𝑛~ \text{for}~\text{ all}~ 𝑥 ∈ [0, 1].$ is closed.

Let $$𝐢[0,1] = \{ 𝑓: [0, 1] β†’ ℝ ∢ 𝑓 ~\text{is}~\text{ continuous}\}$$ and $$d_{∞}(𝑓, 𝑔) = sup\{ |𝑓(π‘₯) βˆ’ 𝑔(π‘₯)|: π‘₯ ∈ [0, 1]\} $$for $𝑓, 𝑔 ∈ 𝐢[0,1]. ~\text{For}~\text{ each}~ 𝑛 ∈ β„•$, define ...
Ricci Ten's user avatar
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3 votes
0 answers
37 views

$L_2$ convergence of bivariate function

I have the following problem: Let $X,Y$ be random variables with distributions $P_X,P_Y$ and $f_0$ be a map from the support of X,Y to the reals. I define a new function $\chi_0(y) = E_X[f_0(X,y)]$. ...
xcesc's user avatar
  • 33
0 votes
0 answers
21 views

Uniformly Continuous and locally Lipschitz but not Globally Lipschitz Function on a "Connected but not compact" set

I know such a function exists but I can’t find an example. I have the famous example f:]0,inf[ --->R f(x)=sqrt(x) function. But I can't find any other function which is Uniformly Continuous and ...
máthΔ“ma's user avatar
2 votes
0 answers
28 views

Attempt to derive Taylor expansion using discrete time steps

I was playing around trying to check if the nth order Taylor approximation about 0 can be explained and obtained by using a physical reasoning argument like below for the case $n = 2$ (that is given $...
Alejandro's user avatar
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0 answers
36 views

Prove that the sequence defined by recurrence is periodic [duplicate]

Been stuck here for a while, can anybody help me? Thanks! Prove that the sequence defined recursively such that $$a_{n+1} = \frac{2}{2-a_n}$$ is periodic and has a period p = 4. We know that $a_1 \neq ...
Mircea Bodean's user avatar
1 vote
1 answer
54 views

Is it possible to show this Integral identity, by assuming the hyposeses I have made?

Assume there exists $p>2$ such that $B\in L^p_{\rm loc}(\mathbb{R}^2)$. Assume in addition that there exists $\tau >2$ such that \begin{equation} \label{B-decay-cond} |B(x)| \, = \, O(|x|^{-\...
X-man's user avatar
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0 answers
65 views

How does Axler know he has found the infimum?

I am reading the following example from Measure, Integration & Real Analysis by Sheldon Axler about the outer measure: Suppose $\displaystyle A=\{ a_1,a_2,...,a_n \}$ is a finite set of real ...
Alice's user avatar
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0 answers
36 views

Proof that the Hamming distance is a metric [duplicate]

If $x$, $y$, are words of length $n$ in a code $C$, $x=x_1 \cdots x_n$, $y=y_1 \cdots y_n$ we define $$d(x, y)= d(x_1, y_1) + \cdots + d(x_n, y_n)$$ from where $$\quad d(x_i, y_i)=0 \text{ if } x_i=...
Wrloord's user avatar
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1 vote
1 answer
62 views

Prove that $c \sup A = \sup(cA)$ for $c>0$.

I'm new to real analysis and trying to prove $\sup⁑(cA)=c\sup⁑(A)$ for $c>0$. Using this definition of least upper bound: $s=\sup A$, where $s\in \mathbb R$ and $A\subseteq \mathbb R$ if $\forall ...
Ba_nanza's user avatar
  • 138
3 votes
0 answers
162 views

Closed form for $\sum_{k=n}^{2n-1} k(-1)^k \sum_{i=k}^{2n-1} {2n \choose i+1} {i \choose n}$?

I've found this sum: $$S(n) = \sum_{k=n}^{2n-1} k(-1)^k \sum_{i=k}^{2n-1} {2n \choose i+1} {i \choose n}$$ The inner sum is elliptical iirc, but perhaps the double sum has a nice expression. We can ...
hellofriends's user avatar
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1 vote
3 answers
98 views

Show that the linear functional is unbounded in $C_{00}$. defined as $T$ is defined as $T(x)=\sum_{n=1}^{\infty} \frac{x_n}{\sqrt{n}},$

Given a linear functional $T: C_{00}\to C_{00}$. Where $C_{00}$ is space sequences with finitely many non-zero terms with $\ell_2$ norm. $T$ is defined as $$T(x)=\sum_{n=1}^{\infty} \frac{x_n}{\sqrt{n}...
Ricci Ten's user avatar
  • 520
1 vote
1 answer
42 views

How to write this in a rigorous way? [duplicate]

I was studying for a test and encountered the following problem in a textbook: Suppose $f(x)$ is continuous, with $f >0$ for all $x$ and $\lim_{x \to \infty} f(x) = \lim_{x \to -\infty}f(x)= 0$ ...
Artur Stolf's user avatar
1 vote
0 answers
42 views

A simple clarification on convergence of functions

Definition: $\lim_\limits{\large x\ \to\ x_0 \atop \large x\ \in\ E}f(x) = L$ iff for every $\epsilon > 0$, there exists a $\delta > 0$ such that $\vert f(x) - L \vert \leq \epsilon$ for all $x ...
Community_Digest's user avatar
1 vote
1 answer
23 views

A function whose points of discontinuity form an arbitrarygiven $F_\sigma$ set.

From 'Counterexamples in Analysis' by Bernard R. GelbaumJohn ,M. H. Olmsted i studied this: The part underlined in red is unclear to me. If $c\in (A_n\setminus A_{n-1})\setminus I(A_n\setminus A_{n-1})...
user791759's user avatar
0 votes
1 answer
27 views

Lebesgue outer measure in $\mathbb{R}^2$ in terms of a grid of $h$-squares

For a set $D\subseteq\mathbb{R}^2$, the Lebesgue outer measure of $D$ is defined by $$\lambda^\ast(D)=\inf\bigg\{\sum_i\lambda(I_i)\mid D\subseteq\bigcup_iI_i\bigg\},$$ where $\{I_i\}$ is a sequence ...
ashpool's user avatar
  • 7,006
0 votes
0 answers
23 views

How to understand that both the lim inf and lim sup of the sets of real numbers is either open or closed?

I am reading the Real Analyais (4th edition) by Royden, H. L., & Fitzpatrick, P. In Section 1.4, it is said that both the lim inf and lim sup of a countable collection of sets of real numbers, ...
Lau's user avatar
  • 11
1 vote
1 answer
39 views

Composition of functions, once for all

I need certainties. Consider two functions $f: A \to B$ and $g: C \to D$. For what I am going to ask, it's not a loss of generality if we consider multivariable functions with domains and or codomains ...
J.N.'s user avatar
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3 votes
0 answers
57 views

Solving for an Exercise from Zorich's Mathematical Analysis I Chapter 5 [closed]

Let $f\in C^{(n)}((-1,1))$ and $\sup_{-1<x<1}|f(x)|\le 1.$ Let $m_k(I)=\inf_{x\in I}|f^{(k)}(x)|$ where $I$ is an interval contained in $(-1,1)$. Show that: a) if $I$ is partitioned into three ...
Yinuo An's user avatar
  • 370
2 votes
1 answer
90 views

We get $L^2$ convergence, but do we get a.s. convergence?

Assume you have a sequence of independent Bernoulli random variables $X_i$ each with probability $p_i$. Let $c_i$ be a sequence of real numbers and $ m,M$ be a real numbers such that $0 < m <c_i&...
user394334's user avatar
  • 1,262
1 vote
1 answer
44 views

Convergence of linear functionals

Let $(\ell_n)_{n\in\mathbb{N}}$ be a sequence of linear functionals in $\mathrm{BV}^*$, namely the dual of the space of functions with bounded variation. Suppose that $\ell_n$ converges to $\ell$ as $...
JayP's user avatar
  • 1,136
1 vote
0 answers
30 views

Question on the formulation of Stokes' theorem

In the book I'm reading, the Stokes' Theorem is stated as follows: Let $\Omega\subset\mathbb{R}^2$ be open, bounded and of class $C^1$. Let $V\subset\mathbb{R}^3$ be an open set. Let $f:V\rightarrow\...
Gary Faust's user avatar
0 votes
1 answer
39 views

Can a function over a domain with holes have a derivative of zero everywhere but fail to be constant? [duplicate]

I've been reviewing some real analysis to prepare for topology and I got myself into a pickle while trying to understand disconnected sets and holes. Here is some reasoning that arrives at a weird ...
Cole Hillyer's user avatar
0 votes
1 answer
31 views

If $(X, A, m)$ is $\sigma$-finite and $B\subseteq A$ is a $\sigma$-sub-algebra, then is $(X, B, \nu)$ $\sigma$-finite?

Here $\nu$ is defined in the following way: $$\nu(E)= \int_E f dm$$ where $f$ is non-negative, $A$-measureable. I don't see why we can assert that $\nu$ is also $\sigma$-finite since the subsets of $B$...
Hyperbolic Cake's user avatar
0 votes
0 answers
86 views

Proof that the $\Bbb Q$ is dense in $\Bbb R$ (need check on intuition) [duplicate]

In this exercise, I am practising by trying to show that $\Bbb Q$ is dense in $\Bbb R$ I begin by stating I want to show that for any epsilon and for any $x$ in $\Bbb R$, there exists a rational ...
another_bik's user avatar
0 votes
1 answer
38 views

A simple question about bounding a sum

Let $\Lambda_N:=\{-N+1,\dots,N-1\}^2\subset\mathbb{Z}^2$ be a set of lattice points in $\mathbb{Z}^2,$ and let $\gamma=\frac{1}{\sqrt{N}}.$ For $x\in\Lambda_N$ with $\|x\|_2>\gamma^{-1}$(with $\|\...
Chang's user avatar
  • 329
4 votes
2 answers
112 views

Global existence of solution of ODE - Gray-Scott model

I am dealing with the ODE version of Gray-Scott model: \begin{equation} \begin{split} \dot{x} &= -xy^2 + F(1-x)\\ \dot{y} &= xy^2 - (F + k)y, \end{split} \end{equation} where $F>0$ and $k&...
Curious's user avatar
  • 55
1 vote
0 answers
55 views

Application of Gronwall's lemma to the expectation

Let $X_t$, $Y_t$ be real-valued continuous stochastic processes with finite second moments such that $$ E |X_t-Y_t|^2 \leq \int_0^t K(s) \left( E |X_s - Y_s |^2 + W_2^2( \mu_s, \nu_s) \right) ds, \...
Holden's user avatar
  • 1,557
1 vote
0 answers
19 views

Conditions on coefficients for a family of multivariate trigonometric polynomials

Let $f$ be a real trigonometric polynomial of $d$ variables that is bounded $|f|\le 1$. Further assume that the maximum degree for each variable is 1. Then we can write it as the sum of monomials, ...
efontana's user avatar
0 votes
0 answers
68 views

Theorem 7.48 in Apostol's MATHEMATICAL ANALYSIS, 2nd ed: Lebesgue's Criterion for Riemann Integrability [closed]

Here is Theorem 7.48 (Lebesgue's Criterion for Riemann Integrability) in the book Mathematical Analysis - A Modern Approach to Advanced Calculus by Tom M. Apostol, 2nd edition: Let $f$ be defined and ...
Saaqib Mahmood's user avatar
-1 votes
0 answers
32 views

Example of the function $F(x)$ that has non zero total variation as $x$ goes to $-\infty$

I am trying to find an example of the function $F(x)$ that has non zero total variation as $x$ goes to $-\infty$. Let $T_F(x) := sup\{ \sum_{i=1}^n |F(x_j) - F(x_{j-1})| :n \in \mathbb{N}, -\infty<...
someeed's user avatar
  • 469
0 votes
0 answers
35 views

Average cluster size of a nxn matrix

I asked a question about the cluster size inside a vector here. As a result, I finally used the expression $\frac{n}{-k+n+1}$ as the average cluster size, although itΒ΄s not proved correct for every ...
Cardstdani's user avatar
0 votes
1 answer
38 views

Compactess of a set in $\mathbb{R}^d$ defined as the union of compact sets

Let $f:[a,b]\to \mathbb{R}^d$ be a function of class $C^1$. Let us consider the following set: $$ \mathcal{A}:=\bigcup_{x\in [a,b]}\{y\in \mathbb{R}^d, \quad ||y- f(x)||\leq 1/2 \} $$ I think that ...
hanava331's user avatar
  • 109
2 votes
0 answers
65 views

How to prove $\int_0^{\infty}e^{-sx}\frac{\sin(x)^2}{x}dx = \frac{1}{4}\ln(1- \frac{4}{s^2})$ using Tonelli-Fubini

I need some help with in trying to prove that $$ \int_{0}^{\infty}{\rm e}^{-sx}\,\frac{\sin^{2}(x)}{x}\,{\rm d}x = \frac{1}{4}\ln\left(1- \frac{4}{s^{2}}\right) \quad\mbox{using}\ Tonelli\mbox{--}...
UDAC's user avatar
  • 597
2 votes
0 answers
51 views

How to define double factorial for non positive integers?

I studied double factorial which known for natural number $$ n!!=n(n-2)!! , 1!!=0!!=1$$ So we have for $n\in N$ $$ (2n)!!=2^n n! , (2n+1)!!=\frac{(2n+1)!}{2^n n!}$$ but I found on Math-World formula ...
Faoler's user avatar
  • 1,637
0 votes
1 answer
50 views

Finding $\delta$ for convergence of $f(x)=x^3+2x^2-3x-1$ at its approximate root.

I've had some exposure to elementary analysis and I am currently going through a problem in numerical analysis involving finding roots using Newton's method. The algorithm has convergence issues which ...
Mario Figueroa's user avatar
3 votes
1 answer
45 views

Distance of a real number to a discrete set of scaled sine values

Let $M>0$ be an integer, $c\in(0,\frac{1}{2})$ a real number, $$ a_{m,n}:=\frac{2n}{\pi}\sin\frac{m\pi}{2n}, $$ $$ A_n:=\left\{a_{m,n}:~m=1,\ldots,n-1\right\}, \text{ and} $$ $$ d_n:=\operatorname{...
Hui Zhang's user avatar
  • 596
1 vote
0 answers
33 views

Application of Stone-Weierstrass theorem to linear combinations of real exponential functions

Theorem 7.32 in Rudin's PMA gives the Stone-Weierstrass theorem as follows: 7.32 Theorem: Let $A$ be an algebra of real continuous functions on a compact set $K$. If $A$ separates points in $K$ and ...
zooond's user avatar
  • 69
0 votes
1 answer
52 views

$C^1[a,b]$ non-linear function, such that $f'' =0$ a.e.

I've been trying to come up with an example of such a function, but haven't succeeded. I was thinking of using a cantor function, since it is continuous, and its derivative is zero almost everywhere......
Hyperbolic Cake's user avatar
2 votes
0 answers
146 views

Prove $\frac{1}{1+f(x)}$ is not integrable over $\mathbb{R}$. [duplicate]

This is a ISI PHD ENTRANCE sample question 2012. The problem Given a non-negative measurable function over $\mathbb{R}$ such that $$ \int_{-\infty}^{\infty}{f(x)}dx=1, $$ then show that $$\int_{-\...
Ricci Ten's user avatar
  • 520
-1 votes
0 answers
72 views

Show that $\liminf(x_k \cdot y_k) = x \cdot \liminf(y_k)$ [closed]

We have a sequence $(x_k)$ in which $\lim(x_k)= x$ and $x > 0$. We also have that $\exists k_0$ such that $y_k > 0 \space \forall k \geq k_0$, then show that $\liminf(x_k y_k) = x \liminf(y_k)$. ...
cardinalcat27's user avatar
0 votes
0 answers
31 views

Extension of a Hölder function on the product space

Let $k\geq0, \alpha>0$ and let $\Omega\subset \mathbb{R}^d$ be a $C^{k,\alpha}$ convex domain. Suppose $f:[0,1]\times \overline{\Omega} \to \mathbb{R}$ satisfies the following: For each $t\in[0,1]$...
Stephen_lamb's user avatar
2 votes
1 answer
65 views

Analytic way to find the sup of this sequence

I'm puzzled on how to find the sup of the sequence $a_n = n\cdot 2^n - n!$, without using tables of values and numerical calculations. Let me explain: by plotting the sequence, or by trying some ...
J.N.'s user avatar
  • 75
3 votes
1 answer
42 views

Why do we need the right-continuity of $F$ at $a$ to prove $V_F[a,c]=\lim_{b\to a^+}V_F[b,c]$ where $F$ is of finite variation?

This is a continuation of this post. The book claims the following without proof: Proposition$\quad$ Suppose that $F:\mathbb{R}\to\mathbb{R}$ is of finite variation. Suppose also that $a<c$ and $F$...
Shenron's user avatar
  • 75
2 votes
3 answers
56 views

Proof of $V_F(-\infty,b]=\lim_{a\to-\infty}V_F[a,b]$ where $F$ is of finite variation.

I am reading the proof of the following result: Proposition$\quad$ Suppose that $F:\mathbb{R}\to\mathbb{R}$ is of finite variation. If $b\in\mathbb{R}$, then $$ V_F(-\infty,b]=\lim_{a\to-\infty}V_F[a,...
Shenron's user avatar
  • 75
0 votes
1 answer
35 views

Showing integrability of f+g and additivity of the Darboux integral

I am currently working on the following question from Measure, Integration & Real Analysis by Sheldon Axler: Suppose $f,g:[a,b]\to\mathbb{R}$ are Riemann (Darboux) integrable on $[a,b]$. Prove ...
Alice's user avatar
  • 508
2 votes
2 answers
215 views

Is $\lim_{n\to\infty}\left\{n!\sum_{k=1}^{n!}\frac{1}{k^{\frac{3}{2}}}\right\}$ not convergent?

I need to calculate the limit (if it exist) $$\lim_{n\to\infty}\left\{n!\sum_{k=1}^{n!}\frac{1}{k^{\frac{3}{2}}}\right\}$$ where $\{x\}$ denotes the fractional part of $x$, $n!$ is the factorial of $n$...
Max's user avatar
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