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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.

1
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1answer
14 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 $$f(x) = \begin{cases} 1 \text{ for } x \in \mathbb{Q}\\ 0 \text{ for } x \notin \mathbb{Q} \end{cases}$$ is not integrale on $[0,1]$. Now I'm at the point in the book where we ...
0
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0answers
11 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)^n) $
0
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0answers
13 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 ...
0
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0answers
18 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
81 views

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

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
26 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
13 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
66 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
81 views

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

How do I solve $\int_0^1(x-1)^2(1-x)dx$?
0
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1answer
11 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
18 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
47 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
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1answer
20 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: $\...
0
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0answers
17 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
35 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(...
-6
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0answers
30 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, ...
0
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2answers
35 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 ...
1
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2answers
41 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
33 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
253 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
51 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
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0answers
39 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
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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
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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
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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
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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
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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
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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
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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
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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
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1answer
57 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
47 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
59 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 ...
1
vote
2answers
22 views

Linearity property of integral involving a bounded linear operator

Suppose $X$ is a Hilbert space and $T\in\mathcal{B}(X)$. For $f\in\mathcal{C}([a,b],X)$, where $a\leq b$, we have \begin{equation} Tf\in\mathcal{C}([a,b],X)\quad\text{and}\quad T\int_{a}^{b}f(x)\,\...
0
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0answers
9 views

Set of roots Hermite polynomials(probabilistic type)

What is the nature of the set of all roots of the Hermite polynomials? They’re known to be all real. Is it a dense set? If not, what are the limit points?Are the limit points also roots? Are the limit ...
1
vote
1answer
41 views

What is $\int_0 ^1 \int_x^1 \frac{f(t)}{t} dt dx $ if $\int f =1$?

Let $f$ be a Lebesgue integrable function on $[0,1]$ and $\int f = 1,$ and let $$g(x) = \int_x ^1 \frac{f(t)}{t} dt \quad x \in [0,1].$$ Calculate the integral of $g$. I feel like I'm supposed to ...
1
vote
0answers
15 views

Sequence of bounded variation functions that converge uniformly to a function with unbounded variation

I need help with this exercise: Prove that $\exists (f_{n})_{n=1}^{\infty}\subset BV([0,1])$ such that $f_{n}\to f$ uniformly, but $\|f_{n}-f\|_{BV}\not\to 0$. So I proposed the sequence $(f_{n})_{n=...
5
votes
1answer
313 views

What does it mean for something to be strictly less than $\epsilon$ for an arbitrary $\epsilon$?

Perhaps a trivial question, but something I never completely understood. If we have shown that $a-b < \epsilon$ for all $\epsilon > 0$, then does that imply that $a-b \le 0$? I"m interested in ...
0
votes
1answer
14 views

Radius of convergence comparison for power series

Given two power series $g (x)=\sum a_n x^n $ and $h (x)=\sum b_n x^n$ with radius of convergence $R_1$ and $R_2$ such that $g(x) \leq h (x)$ for all $x \in \{|x| \leq min \{R_1,R_2\}\}$, Does this ...
0
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0answers
15 views

The behaviour of functions nested under themselves outwith their domains

This question follows from some interesting observations on a sum of reciprocals. Instead of summing them however, we will place each fraction to make a continued fraction. Some visualisations on ...
1
vote
4answers
44 views

How to calculate definite integral of absolute value? [on hold]

How to calculate definite integral of absolute value? How to calculate $\int_0^2|x-1|dx$ for example?
0
votes
1answer
28 views

Proving sequence converges to a Riemann-Stieltjes integral

Consider the following question: Let $f \in \operatorname{RS}_a^b (\alpha)$, that is, the function $f$ is Riemann-Stieltjes integrable over the interval $[a,b]$ with respect to $\alpha = \alpha(...
5
votes
1answer
57 views

Convergence of harmonic functions in $L^1$ implies uniform convergence on compact sets

Resorting to an analog of what's done here, I'm trying to prove the following statement: Let $u_m: \mathbb{R}^n \to \mathbb{R}$ be a sequence of harmonic functions and suppose there exists a ...
0
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0answers
30 views

Test if a function is continuous or has at least one discontinuous vertical asymptote between an interval

Imagine evaluating a function with little intervals incrementally across a graph and testing by using the end points of the each interval (and maybe a midpoint), whether the function is continuous for ...