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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.

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0answers
6 views

What's wrong with this distributional laplacian?

Let $f$ be a locally integrable function on an open set $G$ of $\mathbb{R}^{n}$ and $ n\geq2 $. Suppose $\theta$ is in $ C^{\infty}(G) $ such that its laplacian $ \Delta \theta=1 $ everywhere in $\...
2
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1answer
23 views

Using proof by contradiction to prove Dini's theorem

Does this proof sound right? Thanks Dini's Theorem: If $f$ and $f_n$ are continuous functions on $[a,b]$ such that $f_n \leq f_{n+1} \forall n \geq 1$ and $(f_n)$ converges to $f$ pointwise, then $(...
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0answers
22 views

Extremal value theorem

My lecturer explained the concept of EVT quite poorly and so I am left quite confused - could someone clear up what exactly it is and how it can be used in the real analysis course I’m taking? So far ...
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1answer
27 views

Prove that this series of functions converges on the interval [1,2].

Given $f_n(x) = \frac{x}{(1+x)^n}$, prove that the series $\sum_{n=1}^{\infty} \frac{x}{(1+x)^n}$ converges on the interval $[1,2]$. This was an assignment question that I couldn't get. The professor ...
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1answer
28 views

All bounded measurable functions with $\int_1^2 x^n f(x) dx=0, n=0,1,2…$?

Find all real valued bounded measurable functions $f(x)$ such that $\int_1^2 x^n f(x) dx=0$ for $n=0,1,2...$.
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2answers
54 views

Can the norm of a vector be $\infty$?

I am reading Pugh's Analysis and he defines a norm as a certain type of function from $V \to \mathbb{R}$. However, if we have two normed vector spaces, he later says that we can define the operator ...
1
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0answers
21 views

Constructing a sequence with the following property

Let $\beta_m\searrow 0$ be strictly decreasing, but with the rate and gaps arbitrary. Now I want to construct a sequence $\eta_m$ which also strictly decreases to $0$, with the following properties: $...
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2answers
36 views

Calculating a $\lim_{x\to a} f(x)$ when there is no neighbourhood of $a$ on which $f$ would be defined.

Does it make sense to calculate a limit of a function at some point $a$, if there is no neighbourhood of $a$ on which $f$ would be defined? For example $$\lim_{x\to 0}x\ln{x}$$ doesn't exist, because ...
2
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1answer
35 views

Let $f_n:[1,\infty) \rightarrow \mathbb{R}$ be defined by $f_n(x)=\frac{1}{x}\chi_{[n,\infty)}(x)$. Does $\int f_n \to \int f$?

Let $f_n:[1,\infty) \rightarrow \mathbb{R}$ be defined by $f_n(x)=\frac{1}{x}\chi_{[n,\infty)}(x)$. Does $\int f_n \to \int f$? I know it is an application of DCT but I really don't know to come up ...
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5answers
131 views

What is a “linear function” in the context of multivariable calculus?

On Wikipedia, it says When $f$ is a function from an open subset of $\mathbb{R}^n$ to $\mathbb{R}^m$, then the directional derivative of $f$ in a chosen direction is the best linear approximation ...
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1answer
36 views

Finding where a function converges pointwise and uniformly

Consider $$\sum_{n=0}^{\infty} x(1 - x)^{2n} $$ Where does the sum converge pointwise and uniformly? I think $[0, 2)$ pointwise and $(0, 2)$ uniformly because it becomes a geometric series. Also ...
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1answer
61 views

Is Rudin being redundant in this proof?

My question is about Definition 3.16 and Theorem 3.17 in Baby Rudin. Definition 3.16: Let $\{ s_n \}$ be a sequence of real numbers. Let $E$ be the set of numbers $x$ (in the extended real number ...
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1answer
13 views

Isolated points and limit points

I’m reading a complex analysis textbook, and they do a brief review on real analysis. I’ve attached a screenshot of the page, and highlighted the statement in question. Here is the statement: “Clearly,...
2
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2answers
45 views

Is $\sum _{n=1} ^ \infty \frac{1}{(x+\pi)^2 \cdot n^2 }$ uniformly convergent on $(-\pi , \pi)$?

Is this series uniformly convergent on $(-\pi , \pi)$: $$\sum _{n=1} ^ \infty \frac{1}{(x+\pi)^2 \cdot n^2 }\,?$$ My Attempt: If the series were convergent we would have got a natural number $k$ ...
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0answers
36 views

How to find the irrational roots of a 4th degree polynomial

Please help me to solve this 4th-degree polynomial which may have irrational roots. I failed to solve it using the rational zeros' method. Is there any way to solve this problem? Here is the ...
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0answers
24 views

If $V:\mathbb{R}^2\rightarrow\mathbb{R}$ is $C^1$ and positive definite does its level curves form a continuum around the origin?

Let $V:\mathbb{R}^2\rightarrow\mathbb{R}$ be continuously differentiable and positive definite. I'm currently reading a book where the authors claim that $V$ has a continuum of closed level curves ...
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0answers
46 views

Are functions like $\frac{2\cos(x-\frac{1}{x})}{x^2+2}$ integrable in any way?

Is there a way to integrate $$\frac{2\cos(x-\frac{1}{x})}{x^2+2},$$ perhaps through methods other than Riemann integration? I was trying to use Glasser's master theorem to integrate $$\frac{\cos(x)}{x^...
2
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0answers
18 views

If a sequence of polynomials converges uniformly to a continuous function on the real line, then this function is a polynomial [duplicate]

I'm trying to prove that if $f\colon\mathbb{R}\to\mathbb{R}$ is continuous function and there is a sequence of polynomials $p_n$ that converges uniformly to $f$ on $\mathbb{R}$, then $f$ is a ...
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2answers
37 views

Spectrum of translation operator in $L^2$

From exam preperation. I consider the operator $T f(x) := f(x − 1), x ∈ \mathbb{R}$. First on the space of all function $f:\mathbb{R}\rightarrow \mathbb{C}$. There I found that any number other than $...
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3answers
33 views

Proving a limit exists under some conditions

Suppose that $f(x) > 0$ is integrable and monotone decreasing on $[0, \infty)$. Let $F_{n} = \int_{0}^{n} f(t) \mathop{dt}$, $n = 1, 2, 3, \ldots$. Prove that $$\lim_{n\to\infty} F_{n} $$ ...
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3answers
68 views

$\lim_{n \to \infty}(1+\frac{1}{n^2})(1+\frac{2}{n^2})…(1+\frac{n}{n^2})=e^{\frac{1}{2}}$.

Here is the beginning of a proof: Suppose $0<k \leq n$, $1+\frac{1}{n}<(1+\frac{k}{n^2})(1+\frac{n+1-k}{n^2})=1+\frac{n+1}{n^2}+\frac{k(n+1-k)}{n^4}\leq 1+\frac{1}{n}+\frac{1}{n^2}+\frac{(n+1)^2}...
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2answers
42 views

Does $f$ uniformly continuous on $[0, \infty)$ imply that $\lim_{x\to\infty} f(x)$ exists?

Does $f$ uniformly continuous on $[0, \infty)$ imply that $\lim_{x\to\infty} f(x)$ exists? I think that this is true. Because $f$ u.c. implies $f$ has bounded derivative. Can someone help me confirm ...
3
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2answers
51 views

Prove tht $f(x) = x^{2}$ is uniformly continuous on $\bigcup_{n = 1}^{\infty} [n, n + 1/n^{3}]$.

How can I show that the function $f(x) = x^{2}$ is uniformly continuous on $\bigcup_{n = 1}^{\infty} [n, n + \frac{1}{n^{3}}]$? As $n \to \infty$, I know that we get only all of the integers; ...
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4answers
27 views

Proving a left-hand limit exists

Let $f$ be a bounded and continuous function on $(0, 1)$. Also, suppose $f$ has anti-derivative $F$ on $(0, 1)$. Prove that the quantity $$\lim_{x\to 1-} F(x) $$ exists. I know that ...
1
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3answers
40 views

$\sum a_n b_n$ when $\sum a_n$ convergent and $\{b_n\}$ nonnegative

Let $\sum_{i=0}^\infty a_n$ be a conditionally convergent series, and $\{b_n\}$ be a nonnegative and convergent sequence of real or complex numbers. Does $\sum_{i=0}^\infty a_n b_n$ converge? Do we ...
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1answer
46 views

Show $f(x) = \sum_{n = 1}^{\infty} \frac{\sin(nx)}{2^{n}}$ is infinitely differentiable

I want to show that the function $f(x) = \sum_{n = 1}^{\infty} \frac{\sin(nx)}{2^{n}}$ is infinitely differentiable on $\mathbb{R}$. I have no idea how to do this. I think it's pretty obvious that the ...
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1answer
17 views

Specialized covering lemma for a Hardy-Littlewood maximal inequality

In this question, a suggested approach is given for improving the constant in a Hardy-Littlewood maximal inequality from 3 to 2, and the following lemma is stated without proof: Suppose $K$ is a ...
1
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4answers
46 views

Prove the inequality $(1 + x)^{1/\pi} < 1 + \frac{x}{\pi}$ for $x > 0$

Prove the inequality $(1 + x)^{1/\pi} < 1 + \frac{x}{\pi}$ for $x > 0$ The inequality looks very straightforward to prove; however, I have been struggling for a while. The first thing that came ...
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2answers
24 views

Is $f_{n}$ is analytic on $(a, b)$ and $f_{n} \rightarrow f$ uniformly on $(a, b)$ then is $f$ analytic on $(a, b)$?

Is $f_{n}$ is analytic on $(a, b)$ and $f_{n} \rightarrow f$ uniformly on $(a, b)$ then is $f$ analytic on $(a, b)$? Intuitively, I think that the answer is no. I know that the statement holds for ...
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3answers
33 views

How to sketch this Domain of triple integral

i am having hard time sketching the domain of this : $$ \ \int_0^1\int_0^{1-x^2}\int_0^y f(x,y,z){dz}{dy}{dx} $$ is there an easy way to do that ? i got like cylinder and planes and its hard to ...
6
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3answers
63 views

Example of norm on $\mathbb{R}^2$ that's NOT absolutely monotonic.

We call a norm $\|\cdot\|$ on $\mathbb{R}^n$ absolutely monotonic if $$ |a_i| \leq |b_i|, i=1,\cdots,n \implies \|a\| \leq \|b\|. $$ What's an example of norm on $\mathbb{R}^2$ that's not absolutely ...
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1answer
34 views

Basic property of differential equations

can someone tell me if the following property is true? If $f:]a,b[\rightarrow{]0,\infty[}$ is a continuous function then a maximal solution of the differential equation : $x'=f(x)$ which is defined ...
1
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1answer
26 views

How does this implication work?

I am checking my analysis homework to excercise 20.16 from Ross, Elementary Analysis. How does this implication (see picture) holds/is derived? The context of this question is epsilon-delta definition ...
2
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1answer
35 views

Question about the proof of Stone-Weierstrass theorem (Weierstrass approximation theorem) in Rudin

In Rudin's Principles of Mathematical Analysis, a proof of the Stone-Weierstrass theorem in its original statement is included (3ed, p159): My question is about the step after (51), $P_n(x)=\int_{-1}^...
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1answer
10 views

Sequence of functions who are bounded from below, change of limit and integral

I have a question about changing limit and integral. I know of the monotone convergence theorem, so if my sequence is greater zero and increasing, I can change integral and limit. My question now is, ...
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1answer
30 views

Limit of a series of difference quotients minus derivatives of functions

Let $a,b \in \mathbb{R}_+$ be real positive numbers with and $\frac{1}{2}<a<1$ and let $I=[0,b]$ be a closed real interval Let $\forall n \in \mathbb{N}: f_n(x) : I \to \mathbb{R}$ be the ...
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1answer
14 views

Find partial derivatives of $f(x,y) = F(f_1(x,y), f_2(x,y), f_3(x,y))$ in terms of the partial derivatives of $F.$

We have $f(x,y)= F(\cos(xy), y^2\log(1+x^2),\arctan(x^4+y^2)).$ I think that $f_x = F_x \cdot y\sin(xy)+ F_y\cdot \frac{2xy^2}{1+x^2}+F_z\cdot \frac{4x^3}{1+(x^4+y^2)^2}$ Is this correct?
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2answers
26 views

Proving an inequality related to Riemann integration

Problem Let $f$ be a function such that $|f(u)-f(v)| \leq |u-v|$. Assume that $f$ is integrable on $[a,b]$. Prove that $$\left|\int_{a}^{b} f(x)dx - (b-a)f(c)\right| \leq (b-a)^2/2$$ ...
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2answers
23 views

Analysis - Find a function by using MVT [on hold]

Let $f:R \to R $ be continuous such that $ (xf(x))' = e^{x}$ for $x∈R$ with $x\ne0$. Find a function $f$. Hint : Consider $F(x) = xf(x)-e^x$
1
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1answer
17 views

Can we Combine two convergent infinite series as follows

Given two infinite series $S_0 = \sum_{n=1}^{\infty}\frac{1}{F(x, n)^{p}}$ and $S_1 = \sum_{n=1}^{\infty}\frac{1}{F'(x, n)^{p}}$. Let both of these series be 'convergent' to the same real value $K$ ...
0
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1answer
9 views

Show that $|f(a+tu)-f(a)|\leq \max_{1\leq i\leq n}|f(a\pm te_i )- f(a)|$ where $f:\mathbb{R}^n\to \mathbb{R}$ is function convex.

Show that $|f(a+tu)-f(a)|\leq \max_{1\leq i\leq n}|f(a\pm te_i )- f(a)|$ where $f:\mathbb{R}^n\to \mathbb{R}$ is function convex and $(e_1,e_2,\cdots ,e_n)$ is the canonical basis of $\mathbb{R}^n$ ...
4
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0answers
28 views

An inequality about linear PDE

I am trying to solve the following problem: Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^n, \ n\geq 2$. Let $u\in C^2(\overline{\Omega})$ be a solution of $$\left\{\begin{array}{ll}u_t-...
3
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1answer
55 views

A question about union of non-overlapping intervals and its relation to Arzela's theorem

While trying to answer this question I initially tried to argue via contradiction and was led to the following result: Unproved Theorem: For each positive integer $n$ let $J_n$ be a union of finite ...
1
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1answer
39 views

Let $T(f)(y)=\int_0^\infty K(x,y)\cdot f(x)dx$, $\,$ show $\,$ $\Vert T(f) \Vert _p \le C\cdot \Vert f \Vert _p$

Now here is the full statement. Let $K:(0, +\infty) \times (0, +\infty) \rightarrow \Bbb R$ be a Lebesgue measurable function with $K(kx,ky)=k^{-1}K(x,y)$ for every $k>0$ and let $$\int_0^\...
2
votes
2answers
49 views

Characterization of the sequence $x_1=\cos(x), x_{n+1}=\cos(x_n)$ where $x>0.$

Let $x>0$, $x_1=\cos(x),$ and $x_{n+1}=\cos(x_n), \forall n\geq1.$ Then the sequence $(x_n)_{n\geq 1}$ is bounded but not monotone, bounded but not Cauchy, Cauchy, convergent. It ...
0
votes
2answers
21 views

Lebesgue-Radon-Nikodym decomposition

Please how to find the Lebesgue decomposition of $\nu$ with respect to the Lebesgue measure $m$ where $\nu$ is the Lebesgue-Stieltjes measure associated to the following function: $$ F(x) = \begin{...
2
votes
3answers
41 views

sequence $f(x_n)$ is convergent

I'm preparing for my finals, and I'm completely stuck on this problem: Let $f:(0,\infty)\rightarrow \mathbb R$ be a uniformly continuous function. Show that for any sequence $\{x_n\}$ such that ...
4
votes
2answers
283 views

Proving a set to be countable

A set $S = \left\lbrace \left( x, y \right) \vert x^2 + y^2 = \dfrac{1}{n^2}, \text{ where } n \in \mathbb{N} \text{ and either } x \in \mathbb{Q} \text{ or } y \in \mathbb{Q} \right\rbrace$ is given. ...
3
votes
0answers
97 views

Convergence of $\int_0^\infty \sin(x^m)/x^n dx$

$$\int_0^\infty \frac{\sin (x^m)}{x^n}dx $$ Putting $x^m = t$ $$ \begin{align} \frac{1}{m}\int_0^\infty \frac{\sin t}{t^{(\frac{m+n-1}{m})}}dt \end{align} $$ By applying Dirichlet Test I've been ...
0
votes
0answers
23 views

When the series $\sum_{n=1}^{+\infty }\frac{n^{1000}}{2^{\sqrt[k]{n}}}$ is convergent depending on $k\in \Bbb N$? [on hold]

When the series $\sum_{n=1}^{+\infty }\frac{n^{1000}}{2^{\sqrt[k]{n}}}$ is convergent depending on $k\in \Bbb N$? I don't know how should I start.