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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2
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2answers
34 views

Convergence of vectors in $\mathbb{R}^m$ proof - help understanding a step?

The theorem is as follows: A sequence of vectors in $\mathbb{R}^m $ ($m \in \Bbb Z_+$) converges iff all $m$ sequences of its components converge in $\mathbb{R}^1$. I can follow the (if) direction of ...
0
votes
1answer
41 views

Convergence of Bisection, Secant and Newton's method when there is no root

If there is no root to a function can the bisection, secant or Newton's method still converge to some number e.g. $f(x) = \frac{1}{x}$ on interval $[-1,1]$?
1
vote
1answer
29 views

inverse function theorem: $f$ is invertible(smooth inverse), then Jacobian determinant is not zero?

Source: "An Introduction to Manifolds" by Loring W. Tu, p339, p68 For a proof, see Rudin "Principles of Mathematical Analysis" P221 But, if the map $f$ is invertible(has a $C^\infty$ inverse) in ...
1
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0answers
32 views

Exchange of differentiation and limit on a closed interval [duplicate]

For a differentiable function $f(x,y)$, is the following true,$$\lim_{x\to 0}\frac{\partial f(x,y)}{\partial y} = \frac{\partial \lim_{x\to 0} f(x,y)}{\partial y}$$ on a closed interval $y\in[0,\bar{y}...
0
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1answer
22 views

Can all possible values of this inequality occur? [on hold]

Recall the inequality lim inf $x$ $+$ lim inf $y$ $\leq$ lim inf $(x+y)$ $\leq$ lim sup $(x+y)$ $\leq$ lim sup $x$ $+$ lim sup $y$. By replacing the inequality signs with some equality and/or strict ...
3
votes
0answers
43 views

SOLVED: Interpretation of a coercive function

A continuous function $f:\mathbb{R}^n\to \mathbb{R}$ is called coercive if $$\lim_{\|x\|\to\infty} f(x)=\infty.$$ I'm confused as to how to interpret this limit. I have the 2 following possible ...
2
votes
1answer
58 views

Do Cauchy always converge (for some superset)?

I know Cauchy sequences are only guaranteed converge in complete metric spaces. However, I have been struggling with one issue. It seems that every Cauchy sequence converges, at least in a larger ...
0
votes
1answer
32 views

$\mathcal{M}= \lbrace A \subseteq X | A \: \mbox{or} A^{c} \: \mbox{is numerable} \rbrace$ is a $\sigma$ algebra generated by a singleton family

Let $X \neq \emptyset$ and $$\mathcal{M}= \lbrace A \subseteq X | A \: \mbox{or} A^{c} \: \mbox{is numerable} \rbrace.$$ I want to prove that $\mathcal{M}$ is the $\sigma$-algebra generated by the ...
1
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5answers
69 views

Let $\{a_n\}_{n \in \Bbb N}$ be convergent. Prove/Disprove: $\lim\limits_{n\to\infty}$ $|a_n - a_{2n}| = 0$

My main source of confusion is the use of $a_{2n}$. I’m not sure if this refers to the even entries or something else. I know that since $a_n$ converges then $d(a_n,a) < \varepsilon$, or $\lim\...
2
votes
1answer
30 views

Showing every continuous function $f$ on $[a,b]$ is the uniform limit of a seq of polynomials $(q_{n})$, where $q_{n}(x) = x^{n}p_{n}(x)$

If $0 \notin [a,b]$, show that every continuous function $f$ on $[a,b]$ is the uniform limit of a seq of polynomials $(q_{n})$, where $q_{n}(x) = x^{n}p_{n}(x)$ for polynomials $p_{n}$. There was a ...
1
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2answers
59 views

How can I prove this :$\lim_{x\to \infty } \left(x(x+1) \log \left(\dfrac{x+1}{x} \right)-x\right)=\frac12$ for high school level?

I have tried to evaluate this limit: $$\lim_{x\to \infty } \left(x(x+1) \log \left(\dfrac{x+1}{x} \right)-x\right)=\frac12$$ using $\lim_ {x\to \infty }\left(1+\dfrac{1}{x}\right)^{x}=e$, and using ...
1
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1answer
26 views

Doubt regarding greatest lower bound property implies least upper bound property

Least Upper Bound property - A set $X$ is said to have this property if every non empty subset $A$ of $X$ which is bounded above has the least upper bound. (This does not imply the least upper bound ...
0
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2answers
32 views

What is the mathematical meaning of the following sentence

What is the mathematical meaning of the following sentence: The function $f(x)$ is non-decreasing as $x$ decreases to $0$.
4
votes
2answers
73 views

For what continuous $f$ we can write $f(x)+a\cdot f(x+\alpha)$ as $b\cdot f(x+\beta)$?

After reading about harmonic addition theorem, I am interested in: Find all continuous $f:\mathbb R\mapsto\mathbb R$ that admit identities of the form $$f(x)+a\cdot f(x+\alpha)=b\cdot f(x+\beta)$$ ...
0
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1answer
34 views

Can we deduce that this sequence converges to zero

Let $(x_{k})_{k≥1}$ be a real sequence verifying $x_{k}=0$ infinitely many times. When $x_{k}≠0$, then $x_{k}>0$ and the values of $x_{k}$ decreases to zero when $k$ increases. My question: Can we ...
2
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0answers
18 views

Norm $||.||$ on $C(X)$ is equivalent to $||.||_{\infty}$ norm if evaluation linear functionals on $(C(X),||.||)$ is continuous.

Let $X$ be compact Hausdorff space. Let $||.||$ be a norm on $C(X)$ which makes it into a Banach space and also assume that the linear functional $\lambda_x$ given by $\lambda_x(f)=f(x)$ is continuous ...
1
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2answers
85 views

Uniform convergence of the series $\sum_{n=1}^\infty \frac{1}{n} (\frac{x-1}{x+1})^n$

I'm studying the series $$\sum_{n=1}^\infty \frac{1}{n} \cdot (\frac{x-1}{x+1})^n$$ I have already shown that there is pointwise convergence for $x\in[0, +\infty)$ and total convergence in any ...
6
votes
3answers
64 views

Is it possible to utilize the convergence of the sequence $z_{n+1}=a/(1+z_n)$ to prove that the sequence $x_{n+2} = \sqrt{x_{n+1} x_n}$ is convergent?

I'm doing Problem II.4.6 in textbook Analysis I by Amann/Escher. For $x_0,x_1 \in \mathbb R^+$, the sequence $(x_n)_{n \in \mathbb N}$ defined recursively by $x_{n+2} = \sqrt{x_{n+1} x_n}$ is ...
2
votes
1answer
49 views

Does $\sum_{k=0}^{n-1}\frac{1}{\sqrt{n^2-k^2}}\to \int_0^{1-}\frac{1}{\sqrt{1-x^2}}$?

Consider the sum $$\sum_{k=0}^{n-1}\frac{1}{\sqrt{n^2-k^2}}.$$ Obviously, $$\sum_{k=0}^{n-1}\frac{1}{\sqrt{n^2-k^2}}=\frac{1}{n}\sum_{k=0}^{n-1}\frac{1}{\sqrt{1-\frac{k^2}{n^2}}}.$$ I would like to ...
1
vote
0answers
28 views

A generalized differential equation for a convolved differential operator.

The solution to perhaps the world's very first differential equation $$f'(x) = f(x)$$ is the well known exponential functions $$f(x) = k\exp\left[x\right]$$ But what if we consider another ...
0
votes
1answer
43 views

Does $\sqrt[n]{b_1^n+…+b_m^n}$ converge? [duplicate]

As very first task in our analysis I class we got the following task which gives (only) 1 of 16 points. I add the points in order to calculate the needed effort to solve this task... So let $n,m\...
0
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0answers
36 views

Can the sum $\sum_{n=0}^{\infty} \frac{1}{a^{n(n+1)}}$ be explicitly evaluated?

Can the sum $\sum_{n=0}^{\infty} \frac{1}{a^{n(n+1)}}$ be explicitly evaluated? Here $a>1$. It is clear that the sum converges by Root Test. But how to calculate the value of the sum?
1
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2answers
29 views

How to proceed for the following problem on real analysis

Let $\{a_n\}$ be a recursive sequence given by setting $$a_1>2, \quad a_{n+1}=a_n^2-2 \quad \text{for} \quad n\in \mathbb{N}.$$ Show that $$\sum_{n=1}^{\infty}\frac{1}{a_1a_2\cdots a_n} = \frac{...
0
votes
1answer
32 views

If $X\cap (1-X)=\varnothing$ and $X\cup (1-X)=[0,1]$, can I conclude that $X$ has Lebesgue measure $1/2$?

Let's say I have a set $X\subset[0,1]$, and $1-X:=\{1-x\mid x\in X\}$. Question: If $X\cap (1-X)=\varnothing$ and $X\cup (1-X)=[0,1]$, can I conclude that $X$ has Lebesgue measure $1/2$? I am ...
3
votes
2answers
51 views

Uniform convergence of $f_n(x) = \cos^n(x/ \sqrt{n})$ .

I'm studying the uniform convergence of the following sequence : \begin{equation*} f_n(x) = \left\{ \begin{split} & \ \cos^n\left(\frac{x}{\sqrt{n}}\right) \ \ \textrm{if} \ x \in \left[0, \...
1
vote
1answer
24 views

convergence of the series $\sum_{n=1}^\infty a_n= \frac{(n)(x^n)}{(n+1)^n}$

I have tried to prove the convergence of the series by ratio test and i have got the following form: $\frac{(n+1)(n+1)^n(x)}{(n+2)^n(n+2)(n)}$ And i am getting stuck there, i have no idea how to ...
1
vote
0answers
15 views

Eigenfunction of the Sturm-Liouville equation are non-zero

I am working on the Darboux transformation for the Sturm-Liouville equation $$ - \Psi_{xx} + u(x) \Psi = \lambda \Psi $$ Here we define $ \sigma = \frac{\varphi_x}{\varphi} $, where $ \varphi $ is ...
1
vote
1answer
31 views

Identification of two hypercubes in $\mathbb{R}^d$ [on hold]

Use the notation $k = (k_1,\ldots, k_d) \in \mathbb{R}^d$. How to see that the regions $$ \Big \{ \, \, k \in \mathbb{R}^d \, \, : \sum_{i=1}^d \cos(k_i) > 0 \, \, \Big \} \cap \Big \{ k \in \...
6
votes
2answers
115 views

Can a sequence converges modulo every r>0 but diverge?

Is it possible to have a sequence $\{x_n\}$ of real numbers which diverge to $\infty$ (and has no other finite limit points), but satisfy the condition that $x_n\pmod{r}$ converges for every real $r&...
0
votes
1answer
14 views

is operatory norm weak* - lower semicontinuous?

Let $X$ be a normed space with norm $\|.\|$. We already know that the norm $\| . \|: X \to R$ is weakly lower semicontinuous in the sense, if $x_n \overset{w}{\longrightarrow} x_0$ as $n \to \infty$, ...
1
vote
1answer
18 views

Space of Trigonometric polynomial of degree at most n form subalgebra

I had encountered following in Real analysis by N L Carothers Page 170-172 From this $T_1$ becomes subalgebra . but if we consider $\sin x\in T_1$ , $\sin x.\sin x=\sin^2 x\notin T_1$ so how it ...
4
votes
1answer
39 views

Show uniformly convergent sequence of contraction maps has a fixed point but is not unique?

This question may have appeared in the community but has not been shown yet. Let $(M,d)$ be a compact metric space and for each $n$ in $\mathbb{N}$ let $f_n$ be a contraction mapping. Suppose $(f_n)$ ...
0
votes
1answer
28 views

Intersection of open sets of infinite measure [on hold]

Does the intersection open sets of infinite measure converge to a given set? That is, does $$\bigcap_{k=1}^{\infty} \;\; (k,\infty)$$ converge to a set? Edit: Not asking what the limit of the ...
1
vote
1answer
38 views

Verification: Prove $F : C^1[0,1] \to C^1[0,1]$ is continuous and well-defined.

Verification: Prove $F : C^1[0,1] \to C^1[0,1]$ is continuous and well-defined. Where $F(f(x)) = \sin(x)f(x)$. and $C^1[0,1]$ is equipped with the norm $\lvert\lvert f \rvert\rvert = \sup_{[0,1]} \...
-1
votes
1answer
42 views

What is the taylor series of this function at$ x =0$? [duplicate]

Let the $f(x) =e^{-1 \over x^2}$ for $x \neq 0$ Plus Define $f(0) = 0$ By definition of the differentiation,$ f'(0) =0$ But can't figure out the case of the $f^n(0)$
4
votes
1answer
52 views

When does $e^{hD}$ give a well defined operator on analytic functions?

Let $D$ denote the differentiation operator. It is a classic result that $e^{cD}f(a)=f(a+c)$ if $f(x)$ is an analytic function and the radius of convergence of the Taylor series of $f(x)$ at $x=a$ is ...
-2
votes
1answer
36 views

Convergence of two arbitrary points in compact connected, metric space [on hold]

Let $X$ be a compact connected, metric space. Can we say, for any two-point $x,\,y\in X$ there is a sequence $\{a_i\}$ from $x$ to $y$ such that $a_0=x,\, a_i\in B_{d(x,\,y)}(y),\, \lim_{i\to \infty}...
0
votes
1answer
38 views

The fourier series of periodic and real analytic function

Let $f$ be a real analytic and periodic function defined on the interval $[0, 2\pi]$. Then $f$ is infinitely differentiable for sure. Therefore, the fourier coefficients of $f$ decay faster than any ...
0
votes
1answer
24 views

Formulas for pointwise and uniform convergence

I am looking at formulas for testing pointwise versus uniform convergence from Kenneth Ross' text, which are respectively: $$\lim _{n \rightarrow \infty} f_n (x) = f(x), \ \text{for all } x \in S$$ ...
0
votes
0answers
24 views

Showing inverse of multivariate holomorphic map is holomorphic using real analysis techniques

I'm grading for a graduate real analysis course and looking through the text, trying to do some problems before the term starts so I can stay ahead of things. I found the following exercise in the ...
1
vote
2answers
34 views

Integral of positive Part

Is $$\int_a^b \big(f(x)\big)^+\mathrm{d}x = \left( \int_a^b f(x) \mathrm{d}x \right)^+$$ provided that $f:[a,b]\to\mathbb{R}$ is integrable? This means, can taking positive part and integration be ...
1
vote
3answers
50 views

Prove that $f(1)-f(0)=\int_0^1 f'(t)dt$ using $f(x+h)=f(x)+f'(x)h+o(h)$.

Let $f:[0,1]\to \mathbb R$ a $\mathcal C^1([0,1])$ function. I want to prove that $$f(1)-f(0)=\int_0^1 f'(t)dt$$ using $f(x+h)=f(x)+f'(x)h+o(h)$. In the official solution of my lecture they do as ...
1
vote
1answer
70 views

Proof Correction: if $f = f'$, then $f = e^x$

My textbook states that if $f = f'$, then $f = ce^x$. I can't see the flaw in my proof however (which is totally different from the textbook's). Proof $f$ keeps the same sign. Suppose otherwise, $f$ ...
0
votes
0answers
43 views

Show existence of a solution in differential equation for analytic function

Question: Suppose we have a differential equation $$y'(x) = \phi (y(x))~,$$ where $~\phi~$ is analytic, i.e. $$\phi = \sum_{k=1}^{\infty} a_k x^k\qquad \text{with$~~\qquad$ limsup} |a_k|^{1/k} = 0~.$$...
3
votes
3answers
36 views

How to check if a function is convex

According to a calculus book I have been reading, we call a function $g(x)$ a convex function if $$g(\lambda x +(1-\lambda)y) \leq \lambda g(x) +(1-\lambda)g(y)$$, for all $x,y$ and $0<\lambda&...
2
votes
0answers
59 views

A generalization involving a logarithmic integral

In the preprint, A note presenting the generalization of a special logarithmic integral by Cornel Ioan Valean, it is given the following generalization, Let $n\ge1$ be a positive integer. Then, \...
0
votes
0answers
10 views

Darboux covariance for a higher order linear evolution equation

I am working on Darboux Transformation for equations of the form: \begin{equation} \frac{\partial \Psi}{\partial t} = \sum_{k=0}^N u_k(x,t) \frac{\partial^k \Psi}{\partial x ^k} \end{equation} ...
0
votes
1answer
57 views

Let sequences $a_n,b_n$ be such that $0\leq a_n\leq b_n$. Is it true that $\sum(-1)^n a_n$ is also convergent?

Let sequences $a_n,b_n$ be such that $0\leq a_n\leq b_n$. Also $\displaystyle\sum\limits_{n=1}^{\infty}(-1)^nb_n$ convergent. Is it true that $\sum(-1)^n a_n$ is also convergent? Please give one hint
0
votes
1answer
22 views

Radius of convergence for power series of binomial expansion

Now I know that there are other questions on the radius of convergence for the power series of binomial expansion, but they do not answer my question. I already know that the binomial exapansion is ...
0
votes
2answers
24 views

Doubt in Proof of Boundedness for Continuous Functions

By now, it has been established that the image of a continuous function over a closed bounded interval is also bounded. The proof given below aims to show that the maximum value is attained at some ...