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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.

0
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2answers
28 views

Proof that $b$ is point of closure of $[a,b)$

Let $E=[a,b)$. I want to exhibit that $\{b\}$ is a point of closure of $E$. Attempt at a proof: My goal is to show that for all $\epsilon>0$ there exists a $y\in E$ such that $|b-y|<\epsilon$. ...
2
votes
1answer
33 views

Describe the region where $f_n(x) = nx(1-x^2)^n$ uniformly converges

$f_n(x) = nx(1-x^2)^n$ Describe the region where $f_n(x)$ uniformly converges Try Note : $f_n(x)$ converges pointwise on $(-\sqrt{2}, \sqrt{2})$. I could only prove that, for any interval $I \...
-2
votes
1answer
35 views

What is the integral of $\log\left(x^{2} + k^{2}\right)$ ?.

Where $\displaystyle k$ is a real number ?. I have tried everything but I stuck when I have to find integral of $k^{2}/\left(x^{2} + k^{2}\right)$.
1
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1answer
30 views

Show that if $\inf(A^+)=a>0$ then $a\in A$ and $A=\{za;z\in \mathbb{Z}\}$

$A$ is a set such that $x,y\in A\Rightarrow$ $x-y\in A$, and $A^+$ is a subset of $A$ which contains only its positive elements. I was able to successfully show that if $na\in A$, $n\in\mathbb{Z}, n \...
2
votes
1answer
28 views

confusion about measurability requirements for Lebesgue integral

So, it would appear I have forgotten the basic requirements of Lebesgue integration! Let $(\Omega,\Sigma,\mu)$ be an arbitrary $\sigma$-finite measure space. I'm particularly interested in the real ...
0
votes
2answers
36 views

given $x$ irrational can you find $a,b \in \mathbb{Q}$ such that $a+bx = r$ for all $r \in \mathbb{R}$

given $x$ irrational can you find $a,b \in \mathbb{Q}$ such that $a+bx = r$ for all $r \in \mathbb{R}$. I'm trying to solve this. My attempt consists of choosing $b$ close enough to $bx$ such that $...
0
votes
6answers
78 views

Prove that $\tan\left(\frac{1}{x}\right)$ > $\frac{1}{x}$ in the interval [1,∞) [on hold]

I stumbled upon this while I was solving another question, i.e. the comparison test for proving $\sum_{n=1}^∞$ $\tan\left(\frac{1}{x}\right)$ is a divergent series. I would like a rigorous proof or ...
1
vote
1answer
31 views

Describe the interval where $f_n(x) = \sum_{k=1}^n \frac{x^k}{x^k + 1}$ uniformly converges

Find the interval $f_n(x) = \sum_{k=1}^n \frac{x^k}{x^k + 1}$ uniformly converges. Try First, note that $f_n(x)$ converges pointwise when on $(-1,1)$. I claim, $\forall \epsilon \in (0, 1)$, $f_n(x)...
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0answers
21 views

If $\mu_F\ll\beta_1$ and $\mu_F$ is finite then $F$ is absolutely continuous

If $\mu_F\ll\beta_1$ and $\mu_F$ is finite then $F$ is absolutely continuous. Note: here $\beta_1$ is the Lebesgue measure restricted to the Borel $\sigma$-algebra in the real line. Im not sure if ...
-2
votes
0answers
35 views

Is this function Lebesque-Integrable? [on hold]

Let $\alpha$ be a real integer. We define the mapping $f : \mathbb{R}^{n} \setminus \{0\} \rightarrow \mathbb{R}$ where $f(x) = \alpha \|x\|.$ Show for which $\alpha$ is $f$ Lebesgue-integrable on $...
1
vote
0answers
26 views

Measure of $C^1$ path in $\mathbb{R}^2$

I started studying multivariable integration and still trying to grasp the conecpt of the measure. I`m doing excersices and I keep getting the feeling im doing something wrong so I hope one of you ...
0
votes
3answers
67 views

If $x_1=\sqrt 2$ and $x_{n+1}=(\sqrt2)^{x_n}$ then sequence $x_n$ converges to 2.

If $x_1=\sqrt 2$ and $x_{n+1}=(\sqrt2)^{x_n}$ then show that sequence $x_n$ converges to 2. I know this sequence is monotonically increasing. But how to prove it converges to 2? The sequence is ...
2
votes
2answers
135 views

Differential equation with “backwards product rule”.

If we have the following differential equation, ($h,f$ known, $y$ unknown): $$f'(x)y(x) + f(x)y'(x) = h(x)$$ it would be easy, since we could spot the derivative for a product: $$(f(x)y(x))' = h(x)$...
0
votes
1answer
37 views

The limit of arithmetric mean of a properly divergent sequence

What is the limit of $$ S_n := \frac{a_1+\ldots+a_n}{n} $$ for any positive interger $n$, where $a_n \to \infty$ as $n \to \infty$? I am trying to show that for any $\beta$ there exist $N$ such that ...
1
vote
1answer
23 views

Understanding the solution key to a problem which shows that the integral of a sum equals a given value.

Suppose that the domain of convergence of the power series $\sum_{k=0}^{\infty} c_{k}x^{k}$ contains the interval $(-r, r)$. Define $$f(x) = \sum_{k=0}^{\infty} c_{k}x^{k} \hspace{1cm} \text{ &...
2
votes
1answer
41 views

Twice diffrentiable function $f: \mathbb R \to \mathbb R$ such that $f''(x)+e^xf(x)=0, \forall x \in \mathbb R$ [on hold]

Let $f: \mathbb R \to \mathbb R$ be a twice differentiable function such that $f''(x)+e^xf(x)=0, \forall x \in \mathbb R$. Then is it true that $f(x)$ is bounded in $[0,\infty)$ i.e. does there exist $...
1
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1answer
26 views

How to imply the vanishing gradient condition in KKT?

In Boyd's Convex Optimisation, the following optimisation problem is considered $$ \begin{align} \min\quad & f_0(x)\\ \text{s.t.}\quad & f_i(x)\le 0,\quad i=1:m,\quad m\in\Bbb Z_{\ge 0}\\ &...
-1
votes
1answer
32 views

$A$ is compact and closed then prove …

I am taking an introductory real analysis course and I have difficulty understanding and solving the problem below .Is it trying to say that we have an infimum for the distance between every two ...
1
vote
3answers
89 views

Prove that $\sum_{n=1}^∞\frac{\left(\ln n\right)^3}{n^3}$ is a convergent series by using comparison test

I proved by using the integral test that the series is convergent but can't find a way to prove by using the comparison test, which was required.
1
vote
1answer
29 views

Given $\lim_{n\to\infty} |a_{n}|^{1/n} = \alpha$, prove the following properties.

Given $\lim_{n\to\infty} |a_{n}|^{1/n} = \alpha$, prove: 1) If $\alpha > 0$, show $\sum_{n = 0}^{\infty} a_{n}x^{n}$ converges if $|x| < 1/\alpha$ and diverges if $|x| > 1/\alpha$ ...
0
votes
1answer
28 views

Formula for analytic functions?

In here (third under double infinite series) they list the following formula. $$\displaystyle \sum_{k=0}^{\infty} \sum_{j=0}^{\infty} a_{k,j} = \sum_{j=0}^{\infty} \sum_{k=0}^{j} a_{k, j-k}$$ Is ...
2
votes
1answer
27 views

Existence of polynomial p such that $|f(x) − p(x^2)| < \epsilon$

Let $f$ be a real valued continuous function on $\left[−1, 1\right]$ such that $f(x) = f(−x)$ for all $x \in \left[−1, 1\right]$. Show that for every $\epsilon > 0$ there is a polynomial $p\...
1
vote
0answers
32 views

show that c is a unique fixed point for f over the whole real line R [on hold]

enter image description here I am pretty lost on this stuff, as my professor had us mainly self teach due to lack of time left in the semester. Any help would be appreciated! Thanks
-1
votes
1answer
31 views

Limit of integrable functions is integrable? [on hold]

If $(g_n)$ integrable functions and $\int g_n\to \int g$. Then $\int g<\infty$?
1
vote
1answer
14 views

prove the metric d on $\mathbb{R}^2$ defined by $d((x_1,x_2),(y_1,y_2))=\sqrt{(x_1-x_2)^2+(y_1-y_2)^4}$ satisfy the triangle inequality

I'm a bit stuck with this proof. And also, what's the general strategy I need to bear in mind when proving triangle inequality?
1
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1answer
43 views

For each interval $[a,b]$ contained in $I$, sequence $\{f_{n}:[a,b]\rightarrow\mathbb{R}\}$ converges uniformly to $f : [a,b] \rightarrow\mathbb{R}$

$I$ is an open interval Using the following fact to show this: ${\{f_n\}}$ converges pointwise on $I$ to the function $f$, and ${\{f'_n}\}$ converges uniformly on $I$ to the function $g$ Attempt: ...
-1
votes
0answers
34 views

Does this integral exist? [on hold]

Suppose $ F $ is a bounded closed set in $ \mathbb{R}^{m} $ ($ m>1 $) and $ f $ a measurable function on an open neighborhood of $F$. Suppose for each $ x\in F $ there is a set $ E\subset F $ ...
1
vote
1answer
35 views

Any countable set has measurable zero

To demonstrate that ANY countable set has measure zero, is it sufficient to show that the natural numbers have a measure zero? If so, why; and, if not, why not? Thank you :)
2
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1answer
27 views

Proving the sequence $f_{n} = \sqrt{x^{2} + 1/n}$ converges uniformly to $f(x) = |x|$ on $(-1, 1)$.

I have the following exercise from my book: For each $n \in \mathbb{N}$ and each $x\in (-1, 1),$ define $$f_{n}(x) = \sqrt{x^{2} + \frac{1}{n}}$$ and define $f(x) = |x|$. Prove that the ...
3
votes
0answers
38 views

Convergence of a series involving cosine

Let $x \in (0, 2\pi)$. Is the series $\sum_{n=1}^{\infty} \frac{\cos(n^2x)}{n}$ convergent? My guess is: YES and I would like to use Dirichlet test: however I have troubles proving that the partial ...
2
votes
2answers
40 views

Prove the definition of the arcsin(s).

I am given $\arcsin: S \rightarrow (-\pi/2,\pi/2) $ is the inverse function of sin(t) (restricted to [$-\pi/2,\pi/2$]). I'm trying to prove that $\arcsin(s)$= $\int_{0}^{s}1/\sqrt{1-x^2}$ . My ...
-1
votes
2answers
38 views

How this series a_j converges? [on hold]

We have $a_n\geq 0$ and suppose that $\sum_{j=n}^{2n} a_j\leq \frac{1}{\sqrt{n}}$. I dont know how to derive that $\sum a_j$ converges.
0
votes
1answer
27 views

How to show the existence of a root for a specific equation?

There is an interesting question about the solvability of the following equation. Let $a, b, c, d$ be constant numbers. In addition, these constant real numbers satisfy exponent $a >1$, finite ...
0
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0answers
23 views

Showing Lemma's Fatou for functions not necessarily not negative.

Let $g$ integrable function on $E$ measurable set. Let $(f_n)$ measurable functions and $|f_n|\leq g$ for all $n$. Show that $\int_{E} \liminf f_n\leq \liminf \int_{E} f_n\leq \limsup \int_{E} f_n\leq ...
1
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2answers
28 views

Normed linear space

In Walter Rudin's Complex Analysis, it states that by definition$$\|\Lambda\|=\text{sup}\{\|\Lambda x\|: x\in X, \|x\|\leq1\}$$ and then later he shows that $\|\Lambda x\|\leq \|\Lambda\|\|x\|.$ ...
0
votes
1answer
40 views

Prove that $T$ is a linear operator

Let $(v_n)_{n\geqslant1}$ ∈ $l^2$ be a fixed sequence of real numbers. Define a mapping T on $X=l^\infty$ using the formula $T(a_1,a_2,...) = (v_1a_1,v_2a_2,...),$ $x=(a_1,a_2,...)∈ l^\infty$. I ...
2
votes
1answer
80 views

Show that $f''(x)=0$ for some $x>0$

Let $f: \mathbb{R} \to \mathbb{R}$ with the properties: $f(x)>0, \forall x \geq 0$, $f$ is decreasing and $f'(0)=0$. I want to prove that $f''(x)=0$ for some $x>0$. I have thought the following ...
2
votes
1answer
23 views

Showing that an injective Darboux function is strictly monotone.

I was hoping someone could tell me how to prove the following problem I was given: Let $f:[a,b]\to\mathbb{R}$ be a function such that for every $y\in[f(a),f(b)]$ there exists $x\in[a,b]$ such that $y=...
0
votes
1answer
39 views

Verification that $\lim_{x \to a}\frac{f(x)-f(a)}{x-a}$ and $\lim_{h\to 0}\frac{f(a+h) -f(a)}{h}$ are equivalent definitions of the derivative.

I wanted to verify that for definition of the derivative it is true that: $$ \lim_{x \to a}\frac{f(x)-f(a)}{x-a}= \lim_{h\to 0}\frac{f(a+h) -f(a)}{h}$$ If I denote $h=x-a$, we can let $x\to a$, ...
1
vote
4answers
43 views

Proving that the power series for the cosine function is greater than zero, for $x$ in $[0, \pi/2)$.

I'm trying to prove the cosine power series $$\sum_{k=0}^\infty (-1)^k \frac{x^{2k}}{(2k)!} \;>\;0$$ for all $x \in [0, \pi/2)$. Here, $\pi$ is defined as the smallest positive real such that $...
0
votes
3answers
91 views

Series $\sum(-1)^{a(n)}\frac1{a(n)}$ converging to infinity with $a:\mathbb N\to\mathbb N$ a bijection

I'd love to get a hint or direction on this and figure out a solution myself: Construct a bijection: $$\: f:\mathbb{N} \rightarrow \mathbb{N} $$ such that: $$ \sum_{n=0}^{\infty} -1^{f(n)}\frac{1}{f(...
0
votes
2answers
24 views

Proof that $\tan(x)\leq{}x-\frac{\pi}{4}+\tan(\frac{\pi}{4})$ using the Mean Value Theorem

We're asked to proof the above inequality $\forall{}x\in(\frac{-\pi}{2},\frac{\pi}{4}]$. Although am a bit confused with the fact that $\tan(\frac{-\pi}{2})$ is undefined. And thus am stuck when ...
1
vote
0answers
44 views

Convergence of $b_n = \frac{\sum_{i=1}^na_i}{4}$

Question: Let $\{a_n\}_{n\in \mathbb N}$ be a sequence of real numbers, and for each $n\in \mathbb N$ define $$b_n= \frac{\sum_{i=1}^na_i}{4}.$$ Prove that if $\{a_n\}$ converges to $A$, then so ...
0
votes
1answer
29 views

Show that $|y-y_0|<\text{min}(\frac{|y_0|}{2},\frac{\epsilon|y_0|^2}{2}) \implies \left\vert \frac{1}{y}-\frac{1}{y_0} \right\vert < \epsilon. $

I need to show that if $|y-y_0|<\text{min}(\frac{|y_0|}{2},\frac{\epsilon|y_0|^2}{2})$ and $y\neq 0$ and $y_0\neq 0$ are true, then the following inequality is also true: $$ \left\vert \frac{1}{y}-...
1
vote
0answers
18 views

Trying to Prove a generalization of Raabe's test

I'm trying to prove (or disprove) a statement by Feld: https://math.feld.cvut.cz/mt/txte/2/txe3ea2d.htm It's a generalization of Raabe's test and it says that for a sequence $a_n$, for all n, if $...
0
votes
1answer
18 views

The set of points $p$ such that $|p-p_0|>r$ is open, for any $p_0$ and any $r \ge 0$.

I started the problem off by assuming that $|p-p_0| = r + \delta$ and that $|q-p| < \frac{\delta}{2}$ or equivalently that $q \in B(p, \frac{\delta}{2})$. Then we have that $|q-p_0|=|(p-p_0) +...
0
votes
2answers
42 views

Showing that $\lim_{n \rightarrow \infty} \sum_{k=1}^{\infty}k^{1/2} n^{-1} (1 - \cos(n^2 k^{-2}))=\infty$

Does anyone know how to show $ \lim_{n\rightarrow\infty} \sum_{k=1}^{\infty}k^{1/2} n^{-1} (1 - \cos(n^2 k^{-2}))=\infty$? My thoughts were that you should you a series approximation of some sort, but ...
0
votes
0answers
42 views

Prove $f$ is integrable on $[a,b]$ if $f$ has finitely many accumulation points on $[a,b]$.

Suppose interval $I\in[a,b]$ as finitely many limit points, $f:[a,b]\to \mathbb{R} $ is bounded on $[a,b]$ and continuous on $[a,b]\setminus I$. Use the fact that if $f$ is continuous except at ...
0
votes
0answers
12 views

Finding a Borel-measurable inverse

Let $E$ and $F$ be Borel subsets of $\mathbb{R}$. We say that $m:E\to F$ is measure-preserving (with respect to the Borel measure on $\mathbb{R}$) whenever $B\subseteq F$ is Borel in $\mathbb{R}$ ...
3
votes
2answers
44 views

Finding a limit involving roots without derivatives

I need to find the following limit $$ \lim_{x\to-1}\frac{1+x^{1/7}}{1+x^{1/5}} $$ using no derivatives. I've tried attempting to rationalize or divide by certain polynomials, but nothing has worked. ...