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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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1answer
20 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
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0answers
10 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
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1answer
16 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 ...
0
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0answers
34 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.

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
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1answer
21 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
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1answer
14 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 ...
3
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3answers
57 views

Suppose $x_n$ is a decreasing sequence of positive reals with $\sum x_n$ converges, must $(n\log n)x_n \to 0$

We are able to show, using the Cauchy criterion (using sum from $n$ to $2n$) that $nx_n \to 0$ Explicitly this is $0<nx_{2n}<\displaystyle \sum_{i=n}^{2n}x_n$ and the result follows from squeeze ...
1
vote
1answer
24 views

Can one characterize the set of all $A\subseteq\mathbb{R}$ satisfying $2\cdot A\cdot A\subseteq A$ and $A\cdot(\mathbb{R}\backslash A)\subseteq A$?

This question is a spin-off of this question. When trying to solve that question, we came up with the idea of construction functions using sets $A\subseteq \mathbb{R}$ having the properties that $2xy\...
0
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1answer
29 views

Convergence of a sequence… [duplicate]

Let $\{a_n\}$ be a sequence of real numbers. Define $\sigma_n = 1/n(a_1 + \dots + a_n)$. Suppose that $\lim a_n = a \in \mathbb{R}$. Show that $\lim \sigma_n = a$. Here is my work so far... Fix $\...
0
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1answer
40 views

When is Taylor series a polynomial?

We know that Taylor series is a rational function if and only if its coefficients satisfy the linear recurrence relation. Can we put further conditions on the coefficients so that the Taylor series ...
2
votes
1answer
29 views

A polynomial whose roots form an arithmetic progression

Let $f$ be a fourth degree polynomial whose roots form an arithmetic progression. Prove that $f'$'s roots also form an arithmetic progression. I didn' t make much progress, I just wrote $f(x) =a(x-b-r)...
1
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2answers
46 views

Prove $f(x)= 1$ for $x \in \mathbb{Q}$ and $f(x)=0$ for $x \notin \mathbb{Q}$ is not integrable.

I want to prove that $$f(x) = \begin{cases} 1 \text{ for } x \in \mathbb{Q}\\ 0 \text{ for } x \notin \mathbb{Q} \end{cases}$$ is not integrable on $[0,1]$. Now I'm at the point in the book ...
0
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0answers
27 views

Help calculate the limits

Help calculate the limits: 1) $$\lim_{n\to \infty}\sum_{k=1}^n \frac{1}{\sqrt {k(n-k)}}$$ 2) $$\lim_{n\to \infty}\sum_{k=1}^n \frac{1}{(n-k)\ln{n}} $$ 3) $$\lim_{n\to \infty}{n}^p \sin(\pi(\sqrt 2 ...
1
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0answers
28 views

Application of Hahn-Banach Theorem?

I am going to quote some versions of Hahn-Banach Theorem and try to deduce a statement which might be wrong. Thm 1.1 Let $E$ be a real normed vector space and $F\subset E$ as subspace. $\lambda:F\to ...
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0answers
23 views

Let $f(x)$ be continuously differentiable on $(0,\infty)$ and suppose $\displaystyle\lim_{x\to\infty}f'(x)=0$. [duplicate]

Let $f(x)$ be continuously differentiable on $(0,\infty)$ and suppose $\displaystyle\lim_{x\to\infty}f'(x)=0$. Prove that$\displaystyle\lim_{x\to\infty}\frac{f(x)}{x}=0$. I am trying to use cauchy ...
3
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1answer
101 views

Let $x_{n} = \sqrt{1 +\sqrt{2 + \sqrt{3 + \dots \sqrt{n}}}}$. Show $\lim_{n \rightarrow \infty} x_{n}$ exists. [duplicate]

Let $x_{n} = \sqrt{1 +\sqrt{2 + \sqrt{3 + \dots \sqrt{n}}}}$. show $\lim_{n \rightarrow \infty} x_{n}$ exists. To do this the problem has been broken down into three pieces: a) Show that $x_{n} <...
1
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1answer
30 views

How to prove that $\left(\frac{(-1)^n}{\sqrt{n}}\right)_{n\geq 1}$ is a null sequence?

How to prove that $\left(\frac{(-1)^n}{\sqrt{n}}\right)_{n\geq 1}$ is a null sequence? My attempt via induction: If I prove that the denominator grows faster than the numerator, I can conclude ...
0
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0answers
17 views

Convergence in measure on a finite and infinite measure space

Let $(E,\mathcal{A},\mu)$ be a measure space. Let $(f_n)_{n \in \mathbb{N}}$ be a sequence of measurable functions, which converges in measure to $f:\forall \epsilon>0,\lim_n\mu(\left\{|f_n-f|>\...
0
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2answers
69 views

what is $\lim_{x\to 0} f(x)$

Let $$f(x)=\sum_{n=1}^{\infty}{\sin(nx)\over n^2}$$ Then what is $\lim_{x\to 0} f(x)$ Now I know the series converges uniformly by $M-Test$ (Take $M_n=1/n^2$). What should be my next step. I am ...
-2
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3answers
86 views

How do I solve $\int_0^1(x-1)^2(1-x)dx$? [on hold]

How do I solve $\int_0^1(x-1)^2(1-x)dx$?
0
votes
1answer
12 views

Condition upon integral divergence

I have had no classes in analysis and was wondering if the following is true, and if so, how one proves it. Proposition: Let $f$ be a continuous function along the domain $(a,b)$. Let $f(a)$ be ...
0
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0answers
20 views

Show that for every $\epsilon$ there exists a continuous function $g$ such that $\int_X |f-g|d\mu < \epsilon$.

Let $(X,d)$ be a metric space, and $\mu$ be a Borel measure. Let $f\in \mathscr{L}(\mu)$. Show that for every $\epsilon$ there exists a continuous function $g$ such that $\int_X |f-g|d\mu < \...
-4
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2answers
49 views

Find the sum of the function $\sum_{n=1}^\infty (\cos{x})^n$ on $(0, \pi)$ [on hold]

Find the sum of the function $\sum_{n=1}^\infty (\cos{x})^n$ on $(0, \pi)$ We have $-1\leq\cos{x}\leq 1$. So $(\cos{x})^n \to 0$ as $n\to \infty$ Please solve this problem. Please find the sum.
0
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0answers
30 views

Convergence of complex series $\sum_{n=1}^\infty \frac{e^{i\frac{\pi}{6}n}}{n^\frac{2019}{2018}}$

I have the following series $\sum_{n=1}^\infty \frac{e^{i\frac{\pi}{6}n}}{n^\frac{2019}{2018}}$ and needs to determine whether the series is absolute or conditionally convergent or divergent. I've ...
0
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1answer
10 views

Calculating derivative with multiple variables

Let z = f(x,y), x = x(t,s) and y = y(t,s) all be twice continously differentiable functions Try to find $$\frac{\partial z^2}{\partial t^2}$$ I've tried it and only got: $$\frac{\partial z}{\...
1
vote
1answer
21 views

For a triangular array, if $\max_{1\le k\le n}x_{n,k}\to 0$ and $\sum _{k=1} ^nx_{n,k}\to \lambda$ does $\prod_{k=1}^n(1+x_{n,k })\to e^{\lambda}$?

Consider the following: we have a triangular array of nonnegative numbers $$x_{1,1 } \\x_{2,1 } \ x_{2,2 } \\ x_{3,1 } \ x_{3,1 } \ x_{3,3 } \\... $$ The maximum on each row converges to zero: $\...
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0answers
18 views

Examples of increasing continuous function from $\mathbb{R}_+$ onto $\mathbb{R}_+$ satisfying some inequalities

Let $\varphi : \mathbb{R}_+ \to \mathbb{R}_+$ be an increasing homeomorphism satisfying $\varphi(0)=0,$ where $ \mathbb{R}_+:=[0,\infty).$ For example, $\varphi(s)=\frac{s^3}{1+s^2}$ for $s \in \...
1
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0answers
16 views

prove $ ||u||_{L_2(\Gamma)} \leq (2||v||^2_{L_2(\Omega)} + 2||v||_{L_2(\Omega)} ||\nabla v||_{L^2(\Omega)})^{\frac{1}{2}}$

$$ ||u||_{L_2(\Gamma)} \leq (2||v||^2_{L_2(\Omega)} + 2||v||_{L_2(\Omega)} ||\nabla v||_{L^2(\Omega)})^{\frac{1}{2}}$$ for all $u \in H^1(\Omega)$ for the open unit circle $\Omega$ in $\mathbb{R^2}$. ...
0
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0answers
20 views

Best bound for the remainder of two variables Taylor's theorem

Let $f:\mathbb{R}^2 \to\mathbb{R}$ be a two times differentiable function. Then Taylor theorem says that there exists $t\in [0,1]$ such that \begin{align*} f({\bf x})=f({\bf a})+\sum_{i=1}^{2}\frac{\...
0
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0answers
36 views

Thinking process behind proving $\exists$ solution to $f(x) = x$ when f is bounded

Real Analysis; prove $f(x)=x$ has at least one solution. Here is a solution. I proved initially by doing contradiction, assuming $g(x) = x$ , $\forall x\in$ domain of f, either $f(x) > g(x)$ or $f(...
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0answers
33 views

‎Let ‎$‎c‎>0‎$ ‎and ‎$‎\lim‎ f(x) = L‎$ as ‎$‎x‎\rightarrow ‎+‎\infty‎$‎. Then $‎\lim‎ f(c x) = L‎$ as $‎x‎\rightarrow ‎+‎\infty‎$. [on hold]

‎Let ‎$‎c‎>0‎$ ‎and ‎‎$‎‎‎‎‎\lim‎‎ f(x) = L‎$ as ‎$‎x‎‎\rightarrow ‎+‎\infty‎‎$‎‎‎‎. Then $‎‎‎‎\lim‎ f(c x) = L‎$ as $‎x‎‎\rightarrow ‎+‎\infty‎‎$. Thanks.
0
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1answer
25 views

About Baire's theorem

Exercise $3.22$ of baby Rudin is to prove the Baire's theorem: Suppose $X$ is a nonempty complete metric space, and $\{G_n\}$ is a sequence of dense open subsets of $X$. Prove Baire's theorem, ...
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2answers
36 views

Prove that $f: X \rightarrow R$ is continuous with respect to the metric $d_1$ on $X$ iff it is continuous with respect to the metric $d_2$ on $X$.

Let $X$ be a set and $d_1, d_2$ be metrics on $X$ so that for constants $m,M > 0$ and any $x,y \in X$ we have $md_1(x,y) \leq d_2(x,y) \leq Md_1(x,y)$ Prove that $f: X \rightarrow \mathbb{R}$ is ...
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2answers
44 views

A problem on number of solutions of a functional equations

Find all functions $f:R \rightarrow R$ such that $f(0)=1$ and for all $x\neq -1$ : $f(x)=8f(2x+1)$ (I have found only one solution: $1/(x+1)^3$. Method was by iterated substitution of $2x+1$ ...
2
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1answer
30 views

Compactness when mapping into a higher $L^p$ space and then back

Question: Let $q>p \ge 1$ and let $T:L^p[0,1] \to L^q[0,1]$ be a bounded linear operator. Let $i: L^q[0,1] \to L^p[0,1]$ be the inclusion map (which is bounded). Is the composition $i\circ T$ ...
0
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0answers
34 views

Uniformly Bounded and Bounded Variation

Studying functions of bounded variation, the following exercise came up: Let $(f_n)_{n\in \mathbb{N}}$ be a sequence of functions with $f_n:I \to \mathbb{R}$. Show that: If $(f_n)_{n\in \mathbb{N}}$...
6
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1answer
255 views

Proof verification: At most countably many local maxima

I'd appreciate a second pair of eyes on a proof. I want to prove that a function $f:\mathbb{R}\to\mathbb{R}$ can have at most countably many strict local maxima. The question has been asked elsewhere ...
0
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1answer
53 views

Differentiability of $|x|^p$?

Let $p > 0$, and let $f:\mathbb{R} \rightarrow \mathbb{R}$ be defined piecewise by $f(x)= |x|^p$ if $x \in \mathbb{Q}$ and $f(x)=0$ if $x \in \mathbb{R} \setminus \mathbb{Q}$. For what values of $p$...
1
vote
0answers
40 views

Let $f:[a,b] \rightarrow \mathbb{R}$ is cts with $f(a), f(b) < 0, \int_a^b f(x)dx \ge 0$. Show $\exists c \in (a,b) s.t. \int_c^b f(t) dt \ge 0$.

My solution to this problem is the following: Solution: $[a,b]$ is closed interval, so $f$ is uniformly continuous on the interval $[a,b]$. So, we know that there exists $\delta >0$ such that ...
1
vote
1answer
18 views

Set construction confusion

Let $S$ be an infinite bounded subset of $\mathbb R$. Now let's construct a set $T$ where, $$T=\{x:x \textrm{ exceeds at most a finite number of elements of } S\}.$$ Is there any element of $T$ which ...
2
votes
2answers
39 views

Let $(X,d)$ be a metric space and $ A\subseteq X$ be compact. Prove that for any $y \in X$ there exists $x \in A$ so that $d(y,x) = d(y,A)$ [duplicate]

Let $ (X,d) $ be a metric space and $ A \subseteq X $ be compact. Prove that for any $ y \in X $ there exists $ x \in A $ so that $d(y,x) = d(y,A)$ where $ d(x,y)=|x-y| $. Since A is compact it is ...
0
votes
2answers
36 views

Find the closure and the interior of A

I have to find the closure and the interior of the set A defined by $A =${$(x,\sin(x^{-1})) : x \in R-\{0\}$} $\subset$ $R^2$ I don't know how to start. I know that $\sin(x^{-1})$ has it's maximum ...
0
votes
0answers
8 views

Uniform convergence and integrals when domain is not a compact set.

Suppose sequence of continuous functions ${f_n}$ that converges uniformly to a continuous function $f$ on a closed interval $[a,b]$, then we have $$\lim_{n\to\infty}\int_a^b f_n(x) dx = \int_a^bf(x) ...
0
votes
0answers
31 views

If $\sum a_n$ is convergent but not absolutely, then $\sum a_n^+$ diverges

Let $a_n \in \mathbb{R}$, such that $\sum_{n=1}^\infty|a_n|= \infty$ and $\sum_{n=1}^m a_n \to a$, as $m \to \infty$. Let $a_n^+=\max\{a_n,0\}.$ Show that $\sum_{n=1}^\infty a_n^+= \infty$. Approach: ...
3
votes
3answers
108 views

If $\int_a^bg(x)dx=0$, show that $\int_a^bf(x)g(x)dx=0.$

Let $g(x)\ge0$. If $\int_a^bg(x)dx=0$, show that $\int_a^bf(x)g(x)dx=0,$ where $f$ is any integrable function. If simeone is allowed to use the Mean Value thorem for integrals, the proof is at hand. ...
0
votes
1answer
30 views

Proving Borel measurability of a function.

Suppose $f: \mathbb{R}^2 \to \mathbb{R}$ is a function such that $f(x, \cdot)$ is Borel-measurable for each $x$, and $f(\cdot, y)$ is continuous for every $y$. Define $f_n: \mathbb{R}^2 \to \mathbb{...
1
vote
1answer
41 views

Where did this definition come from?

Let $C,D > 0$. We call a function $f : \Bbb R \to \Bbb R$ pretty if $f$ is a $\Bbb C^2$-class, $|x^3 f(x)| \leq C$ and $|xf''(x)| \leq D$. (i). Show that if $f$ is pretty, then, given $\...
0
votes
1answer
59 views

Why is $\lim_{t\to\infty}\frac{(f(t))^{1+\varepsilon}}{tf'(t)}=\infty$ for all $f:[0,\infty)\to[0,\infty)$ such that $f'>0$? [on hold]

Let $f:[0,\infty)\to[0,\infty)$ be a differentiable function such that $f'(t)>0$ for all $t$ and let $\varepsilon>0$. Why is $$\lim_{t\to\infty}\frac{(f(t))^{1+\varepsilon}}{tf'(t)}=\infty\quad?...
0
votes
1answer
51 views

Is this product strictly positive?

Let $p\in (0,1)$ and $\varepsilon\in (0, 1)$ be fixed. For all $i\in \mathbb N$ we define $p_i=1+(p-1)(1-\varepsilon)^i$. Is it possible to prove that $$\prod_{i=1}^{+\infty}\frac{p_i}{2-p_i}>0$$? ...
0
votes
3answers
64 views

I am stumped on this problem

It is from Intro to Analysis by Bartle 3rd ed Let $I:=[a,b]$ let $f:I\rightarrow\mathbb{R}$ be a continuous function with $f(x)>0$ Prove that there exists a number $c>0$ such that $f(x)\geq c ...