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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.

4
votes
4answers
45 views

Let $x_{n+1} = \frac{1}{2}(x_n + \frac{a}{x_n})$. Prove that $x_{n+1} < x_{n}$

Let $$x_{n+1} = \frac{1}{2}(x_{n} + \frac{a}{x_{n}})$$ Prove that $x_{n+1} < x_{n}$ for $a \geq 0$. Hint: Let the initial guess satisfy $x_{1} > \sqrt{a}$ I am stuck at how to begin this. I ...
1
vote
1answer
42 views

Show $e^{a-b}=\frac{e^a}{e^b}$ for every $a,b \in \mathbb{C}$.

I would appreciate to have my proof verified Show $e^{a-b}=\frac{e^a}{e^b}$ for every $a,b \in \mathbb{C}$. As is already known that $e^{a+b}=e^{a}e^{b}$ for every $a,b \in \mathbb{C}$, then for ...
-5
votes
0answers
29 views

A full detailed proof that this partition of the rationals is the square root of 2 and I need it fast-deadline approaches on a publication!

Alright, I know there's already a posted question on this here. But I'm at the end of my rope and need some help here or I'm going to post a false proof and look like a horse's ass in my published ...
0
votes
4answers
35 views

Derivative of $f(x)=\int_{x}^{\sqrt {x^2+1}} \sin (t^2) dt$

Derivative of $f(x)=\int_{x}^{\sqrt {x^2+1}} \sin (t^2) dt$ Firstly I wanted to calculate $\int \sin (t^2) dt$ and then use $x$ and $\sqrt {x^2+1}$. But this antiderivative not exist so how can I do ...
-2
votes
1answer
27 views

Every isometry is Lipchitz-continuous

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces, $f:(X,d_X) \rightarrow (Y,d_Y)$ be an isometry. Then $f$ is Lipchitz-continuous. Attempt: Suppose that $f$ is an isometry. Then for all $x_1,x_2$ in $...
1
vote
1answer
43 views

Application of Jensen's Inequality to non-negative integrable function

I am reading a book, where it uses the following result. Can someone help me to derive the result? I know i have to use Jensen's inequality here, but not sure how to get the final result. Here is the ...
-1
votes
1answer
17 views

A uniformly continuous function with the supremum metric…

Question: Suppose that $C([0, 1])$ is the metric space of all continuous real-valued functions on $[0, 1]$, with the metric $d(f, g) := \sup_{x \in [0, 1]}|f(x) - g(x)|$. Let $f \in C([0, 1])$ such ...
0
votes
3answers
31 views

Why is it possible to calculate multivariable limits using polar coordinates?

Why is it possible to calculate multivariable limits using polar coordinates? Let's say I'm looking for some $\lim_{(x,y) \to (0,0)}$ and I'm substituting $x = r cos\theta$ and $y = rsin\theta$ so ...
2
votes
1answer
45 views

Multivariable limit of $\lim_{(x,y) \to (0,0)} \frac{2xy^4}{(x^2+y^2)^2}$

How can I calculate the multivariable limit of $\lim_{(x,y) \to (0,0)} \frac{2xy^4}{(x^2+y^2)^2}$? I'm new to this and I've seen a couple of examples, where it is possible to limit the fraction by a ...
1
vote
2answers
59 views

Calculate $\int e^{2x}(\cos x)^3 dx$

Calculate $$\int e^{2x}(\cos x)^3 dx$$ My try: Firsty I tried to use integration by parts but then I got: $$\int e^{2x}\cos^3(x) dx=...=\frac{1}{2}\cos^3(x) e^{2x}+\frac{3}{2}\left(\int e^{2x} \sin(...
1
vote
0answers
35 views

Show that $f(x) = {1\over x^n}$ is continuous in its domain, $n\in\Bbb N$

Let $n\in\Bbb N$. Show that $$ f(x) = {1\over x^n} $$ is continuous in its domain. I've recently shown that $g(x) = x^n$ is continuous everywhere in $\Bbb R$. Now I want to do the same for $1/x^n$,...
0
votes
0answers
16 views

Inverse image of a variable under composite mapping

Let $f: X \to Y$ and $g: Y \to Z$. ($.^{-1}$ denotes inverse image). Is the set $f^{-1}(y)$ equal to the set $(g\circ f)^{-1}(g(y))$ for all $y$ ?
1
vote
1answer
12 views

Show there are basis vectors such that $Dg(a)(e_{j_k})$ spans $\mathbb{R}^p$

Given $p<n$, $g:\mathbb{R}^n\to\mathbb{R}^p$ a $C^1$ function with $g(a)=0$ and $Dg(a): \mathbb{R}^n\to\mathbb{R}^p$ surjective. And using the identification $\mathbb{R}^p\cong \mathbb{R}^p\times \{...
0
votes
1answer
23 views

Directional derivative equivalent definition

Let $f=f(x_1,\dots,x_n)$ be a scalar function defined on some open subset $U\subset\mathbb{R}^n$. Given an unit vector $v=(v_1,\dots,v_n)$ and a point $x_0\in U$, the directional derivative of $f$ at $...
0
votes
1answer
19 views

Total Variation of Subintervals

Studying functions of bounded variation, the following exercise showed up: Let $I = [a,b]$ be an interval, $(E, \| \cdot\|)_E$ a normed vector space and $f \in BV(I, E)$ a function which is ...
0
votes
0answers
16 views

Explanation of Nested Interval Theorem

I have a question relating to the theorem about nested intervals. I understand it until the last expression where it is stated that the interval $[a,b]$ is included in the intersection of those ...
2
votes
3answers
33 views

Partial derivative of the real part of a function

I'm trying to understand the mathematical reasoning behind the example provided in this question. If we have $$z = Ae^{i(\omega _{o}t+\phi )}$$ and define $$x = Re (z),$$ then why is it that $$\...
0
votes
1answer
19 views

Algebraic closure of p-adic rationals, $\overline{\mathbb Q}_p$, and its completion, $\Omega_p$, are not locally compact

Trying to show $\overline{\mathbb Q}_p$ and $\Omega_p$ are not locally compact. I can prove it by showing that the unit sphere is not locally compact. That is to say, any sequence on the unit sphere ...
1
vote
1answer
31 views

number of zeroes of arbitrary function

Sorry if I misused/mixed up some maths terms. I barely know any maths lingo, especially not in English. I was thinking about programmatically solving equations (or rather, approximating their roots), ...
0
votes
1answer
38 views

$f$ is Lipschitz and $X$ have measure zero. Show that $f(X)$ has measure zero too.

If $f[a,b]\to\mathbb{R}$ is Lipchitz and $X\subset [a,b]$ has measure zero show that $f(X)$ has measure zero too. What I did: As $X$ has measure zero, $\forall \epsilon>0$ there exists a ...
2
votes
1answer
43 views

Proof that $f$ is differentiable

Let $$f(x) = \sum_{n=1}^{\infty} \frac{x^n}{2^n} \cos{nx}$$ Proof that $f$ is differentiable on $(-2,2)$ my approach let $ m := \frac{x}{2} $ so $m<1$ $$ \left| \frac{x^n}{2^n} \cos{nx} \right| ...
1
vote
1answer
28 views

A Lipschitz function is $C^1$?

I am wondering if a Lipschitz function $f:[a,b]\to\mathbb{R}$ is $C^1$, that is its derivative is also continuous? I have seen that in a text however I could not prove it and does not seem so obvious ...
-1
votes
1answer
31 views

solution of system

Does the solution of the system satisfy $z=v=0$ and why? \begin{align} -a^{2}z-z_{xx}+i\beta a v&=0 \\ -a^{2}v-v_{xx}-i\beta a z&=0 \end{align} where the system is defined on $(l,L)$, with $...
-1
votes
1answer
26 views

What's the difference between the operator norm and the sup norm

What's the difference between the operator norm and the sup norm over $C[0,1]$. a.k.a $\left\lVert x\right\rVert_\infty$ vs $\left\lVert x\right\rVert_{op}$
-2
votes
0answers
17 views

Problem Concerning Cauchy Principle for Sequences.

I have a question but can't seem to figure out how to solve it. The problem states: Let's consider a sequence $x_n$, such that $x_n\to a$, as $n \to \infty$. Using the Cauchy Principle prove that (a)...
1
vote
0answers
37 views

$(x+y)^r \le x^r+y^r$ when $r \in (0,1)$ and $x,y$ are real positive numbers. [duplicate]

I'm sorry for the silly question, but I have a doubt. Given two positive real numbers $x,y$ and taking $0<r<1$, is it true that $$ (x+y)^r \le x^r+y^r? $$ In all the examples I considered, ...
-2
votes
1answer
30 views

Conditionally convergent series, true or false [on hold]

It is given that the series $\sum_{n=1}^{\infty}a_n$ is convergent but not absolutely convergent and $\sum_{n=1}^{\infty}a_n=0$ denote by $S_k$, the partial sum $\sum_{n=0}^{k}a_n,~k=1,2,...$, then $...
-4
votes
0answers
21 views

Exercise about ordinary differential equation

I have some problem about ordinary differential equation. Please help me solve and explain it. Thank you very much. Good health! Problem 1. Suppose $y(x)$ is a solution of the equation $$y''-2y'+y=2e^...
0
votes
0answers
14 views

mean of a field along the line

What is the proof of a mean field along any line? Or how can we define it? i.e \begin{equation} a=\int_0^1 b \, dx \end{equation} where $a$ is the mean of $b$ and $b(0)=b(1)=0$.
3
votes
2answers
57 views

Show power series converges for every $x$.

Let $$f(x) = 1 + a_{1}x^{1}+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}+...$$ be a solution of the differential equation $f'(x)=xf(x).$ Now I need to explain that the power series that define $f(x)$ converges ...
3
votes
1answer
40 views

Convergence/Divergence speed of $u_n = \int_0^1 f(x)^n g(x) \textrm{d}x$ given $f, g$ continuous and non-negative

Let be $f, g : [0, 1] \to \mathbb{R}_{+}^{*}$ continuous maps such that: $\forall n \in \mathbb{N}, u_n = \int_0^1 f(x)^n g(x) \textrm{d}x$ I want to show that $v = \left(\dfrac{u_{n + 1}}{u_n}\...
0
votes
1answer
45 views

For $a>0$,the series $\sum_{n=1}^\infty a^{\ln n}$ [duplicate]

For $a>0$,the series $\sum_{n=1}^\infty a^{\ln n}$ is convergent if and only if (1). $0<a<e$ (2). $0<a\leq e$ (3). $0<a<\frac{1}{e}$ (4). $0<a\leq \frac{1}{e}$ I tried ...
1
vote
2answers
27 views

For some $c \geq 0$ $\text{sup} \ \{ c \cdot f(x): \text{some domain of $x$} \}$ = $c \cdot \text{sup} \ \{f(x): \text{same domain of $x$} \}$ [duplicate]

How would you formally justify this? Or is it just notationally obvious? (As opposed to 'conceptually' obvious, which is never an excuse in mathematics.) Edit: For some $c \geq 0$ $\text{sup} \ \{ c ...
1
vote
0answers
27 views

Can we write the following limit -equality?

We know: 1-) The following condition is valid ONLY with zeros (any) for $\,\,\,\,\,$ $0<x<2$ and $x\in \mathbb R $: $$f(x_o)=g(x_o)=0 \,\,\,\,\ ⇔ \,\,\,\,\ f(2-x_o)=g(2-x_o)=0$$ 2-) ...
1
vote
0answers
45 views

Let $f\in \mathscr{L}^1$. Show that for every $\epsilon$ there exists a continuous function $g$ such that $\int_X |f-g|d\mu < \epsilon$.

Let $f\in \mathscr{L}^1$. Show that for every $\epsilon$ there exists a continuous function $g$ such that $\int_X |f-g|d\mu < \epsilon$. Since the question is asking for a sequence of continuous ...
3
votes
3answers
35 views

$g(x)=\frac{1}{x}\int_{0}^{x}f(u)du$, which of the following option is true?

Suppose $f$ is an increasing real valued function on $[0,\infty)$ with $f(x)>0$ for all $x$ and let $g(x)=\frac{1}{x}\int_{0}^{x}f(u)du$; $0 < x <\infty$, Then which of the following are true:...
0
votes
1answer
29 views

$\liminf_\limits{n\to\infty}1_{A_n}(x)=1$ $\implies$ $\lim_\limits{n\to\infty}1_{A_n}(x)=1$?

Source: Partial proof from textbook: I've omitted the case where $x\in A^c$ as it's not relevant. I've also highlighted the part I'm having trouble with in blue. Here is my attempt at explaining ...
0
votes
0answers
24 views

what is the difference between $\cup_{n=1}^{\infty}(\frac1 n,1)$ ,$\lim_{n\to\infty} (\frac1 n,1)$, $(0,1)$? [on hold]

what is the difference between $\cup_{n=1}^{\infty}(\frac1 n,1)$ ,$\lim_{n\to\infty} (\frac1 n,1)$, $(0,1)$? Are they just the same thing?
1
vote
1answer
30 views

$[F]_p\le [f]_1[g]_p$ for $1\le p\le\infty$

For real-valued functions $f$ and $g$ on $(0,\infty)$, let $$F(x)=\int_0^\infty f\left(\frac{x}{y}\right)g(y)\frac{dy}{y}$$ If $1\le p\le\infty$, set $$[h]_p=\left(\int_0^\infty |h(x)|^p\frac{dx}{...
-1
votes
0answers
65 views

$\int_{0}^{1}|f'|\leqslant c\int_{0}^{1}(|f|+|f''|)$

Prove that exist $c>0$ such that $\int_{0}^{1}|f'|\leqslant c\int_{0}^{1}(|f|+|f''|)$ for all $f$ $\in$ $C^2(0,1)$. Maybe that'll help, we can use similar statement about supremums: $\sup_{(0,1)}|...
0
votes
1answer
42 views

How much do tails contribute to a Gaussian's total variance?

H${}$ello, if $X\sim \mathcal{N}(0,I_{n\times n})$ what is a good upper bound for $\frac{1}{n}\int_{A} \|X\|^2 d\mathbb{P}$ when $\mathbb{P}(A)<\varepsilon$? Thanks!
0
votes
1answer
26 views

Prove that for any $N \geq 0$ the set $A_N = $ {{$x_n$} $\in A: x_n=0$ for $n \geq N$} is compact.

Define the set $A \subseteq {\ell}^2$ by $A = ${{$x_n$} $\in {\ell}^2 : \sum_{n=0}^{\infty}(1+n){|x_n|}^2 \leq 1$} i) Prove that for any $N \geq 0$ the set $A_N = $ {{$x_n$} $\in A: x_n=0$ for $n \...
-2
votes
0answers
48 views

$\int_{0}^{\infty} \frac{f(x)-f(x+1)}{f(x)}dx=+\infty$ [on hold]

Let $f \colon [0,\infty)\to\mathbb{R}$ be a strictly decreasing, continuous function. Suppose $\lim_{x\to\infty} f(x) =0$. Prove that $\int_{0}^{\infty} \frac{f(x)-f(x+1)}{f(x)}dx=+\infty$.
0
votes
1answer
17 views

Is induction the correct approach here?

Let $a_1, a_2, \dots, a_n$ (real numbers) be such that $$a_1 - a_2/3 + \dots + (-1)^{n - 1}a_n/(2n - 1) = 0.$$ Prove that $$f(x):= a_1\cos(x) + a_2\cos(3x) + \dots + a_n\cos([2n - 1]x) = 0,$$ for ...
0
votes
1answer
48 views

series that diverges

We consider the sequence $$(u_n)_{n\in\mathbb{N}}$$ defined by $0<u_0<1$ and $u_{n+1}=u_n-u_{n}^2$ for all $n\in\mathbb{N}.$ I want to prove that the serie with general term $\ln(\frac{u_{n+1}...
0
votes
0answers
36 views

Is a partial differential equation satisfied after reduction to a subspace?

I have a $n$th-order non-linear partial differential in $m$-real variables $x_1,x_2, \ldots, x_m$. Assume a function $f$ satisfies this differential equation. I denote this by $$D f(x_1, x_2, \ldots, ...
0
votes
1answer
33 views

Struggling with finding a potential counterexample for a convergent series.

This question comes with two parts. Part (a): Let $\{f_n(x)\}$ be a sequence of nonnegative functions for $x \in S \subseteq \mathbb{R}$ such that $f_1 \geq f_2 \geq \dots \geq 0$, and that $f_n \to ...
-1
votes
0answers
64 views

Let $f_n$ is continuous function from $[0,1]$ to $\mathbb{R}$ for any natural $n$. And for any $x$ from $[0, 1]$ series $\sum_n f_n(x)$ converge. [on hold]

Prove that exist a positive length interval $[a, b]$ in $[0, 1]$ such that partial sums of series $f_n$ is evently limited in $[a, b].$
-2
votes
1answer
27 views

I need to know if.my example for this problem solved

Let $f(x)=\sup(x^3, x^2+1)$, $I=[1,4]$ $\sup(x^3)=64,\ \inf(x^3)=1$ $\sup(x^2+1)=17,\ \inf(x^2+1)=2$ So $\sup(x^3,x^2+1)=\sup(64,17) =64$ Is this correct, can I have good example.
0
votes
1answer
22 views

Proof of Density of Rational Numbers as given in Howie

I'm working out of John M. Howie's Real Analysis (2001). In section 1.4 (Exercise 1.5), we're asked the following: Let $x$ and $y$ be real numbers, with $ x < y $. Show that, if $x$ and $y$ are ...