Questions tagged [real-analysis]

For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property and related analysis topics, such as continuity, differentiation, and integration.

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Stein's proof of Lusin's theorem: Is the $E_n$ necessary?

I wonder if this is a simpler and correct proof of the Lusin theorem stated below. I was inspired by Stein's proof. However I think the $E_n$ in Stein's proof may be unnecessary. This is the theorem ...
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2 votes
0 answers
52 views

Prove that $\displaystyle{\lim_{k \to \infty}} k^{n-1} \phi(k) f(kx) \neq 0$

This is a homework problem of my real analysis class. In the previous parts of this problem, I have shown that for a fixed $f \in L^1(\mathbb{R}^n, m^n)$, we have $$\lim_{k \to \infty}k^{n-1}f(kx) = 0$...
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1 answer
65 views

Can I approximate$\int\limits_{0}^{\pi/4} e ^{\tan(x)} \mathrm{d}x$ using Taylor series?

I have a question regarding the following integral, I know that its solution is Complex. But my question is, can I approximate the integral to a real value, using Taylor series? My analysis is the ...
0 votes
0 answers
17 views

Sobolev space on Carnot group with zero trace

Let $\mathbb{G}=(\mathbb{R}^{n},\circ)$ be a Lie group on $\mathbb{R}^n$ and $\mathfrak{g}$ be the corresponding Lie algebra of $\mathbb{G}$. Let $X_{1},\ldots,X_{m}$ be the left-invariant smooth ...
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0 answers
60 views

Derivative of $\frac x {\log |x|}$ without using L'Hopital's rule?

Consider the function $f:x \neq 0 \mapsto \frac x {\log |x|}, 0 \mapsto 0$. What is its derivative at $x = 0$? I was only able to solve this using L'Hopital's rule. I request: Verification or ...
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0 votes
0 answers
52 views

Show $f(x) = |x|^p$ is differentiable at $0$ for $p > 1$

Show $f(x) = |x|^p$ is differentiable at $0$ for $p > 1$. My proof is below. My question is to verify or improve it, or respond to the remark. Proof: We examine the limit from both the right and ...
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1 vote
1 answer
27 views

Question about notation in the proof that if the lim(s_n) is defined, the liminf(s_n) = lim(s_n).

I am reading Elementary Analysis by Kenneth Ross and am really struggling, especially with the choice of notation and why certain things are included in the proofs. In chapter ten he is proving that ...
1 vote
1 answer
52 views

Let $A$ be a non-empty, proper subset of the reals $\mathbb{R}$. What "obvious" set relationship can be proven about $A$, $\bar{A}$, and/or $A^c$.

I'm given a list of test-prep problems, which involve unstated theorems which we are to guess the statement of and then prove. I'm able to work out what most expect me to show, however the following ...
1 vote
3 answers
72 views

Prove that the set is not open

I want to prove that $$\bigcap _{n\in \mathbb{Z}^{+}}\left(-\frac{1}{n}, \frac{1}{n}\right)$$ is not open in $\mathbb{R}$ with usual metric. For this I assumed that it is open, then for $x\in \bigcap ...
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2 votes
1 answer
81 views

If $X_n>0$ and $X_n\uparrow X$ with $E(X)<\infty$, then $E(X_n|\mathcal{F})\uparrow E(X|\mathcal{F})$.

The theorem below is from Durret's Probability Theory and Example. I am struggling to follow the proof of (c). I will try to give all my thought here. By the definition of conditional expectation and (...
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1 vote
3 answers
89 views
+50

Why is the wiener algebra a subset of continuous functions

I am studying Fourier analysis and I am currently reading about the Wiener algebra. The Wikipedia page claims that $A(\mathbb{T}) \subset C(\mathbb{T})$ (see: https://en.wikipedia.org/wiki/...
1 vote
1 answer
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How can you prove that $\lim\limits_{x\rightarrow z}[\lim\limits_{y\rightarrow x}f(y, x)]=\lim\limits_{y\rightarrow z}f(y, z)$?

I want to prove that: $$\lim\limits_{x\rightarrow z}[\lim\limits_{y\rightarrow x}f(y, x)]=\lim\limits_{y\rightarrow z}f(y, z)$$ As long as all the limits exist (and with no other restrictions or ...
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-1 votes
1 answer
28 views

Basis for euclidean topology

Let $B(x,r) \in R^n$ with euclidean metric and let be the following subsets of $R^n$ A={$B(x,r): x \in \mathbb{R^n}, r \in \mathbb{Q^+}$} B={$B(x,r): x \in \mathbb{Q^n}, r \in \mathbb{Q^+}$} C={$B(...
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0 votes
1 answer
60 views

Ways to find coefficient of $x^{2n}$ in $\prod_{i=1} ^\infty (x^2 - \frac{i^2π^2}{4})$

How to find the coefficient of $x^{2n}$ ($n\in N$) in this infinite product ? $$\prod_{i=1} ^\infty (x^2 - \frac{(2i-1)^2π^2}{4})$$ By equating it with the series expansion of $\cos x$ , I know that ...
3 votes
1 answer
60 views

Upper bound for the norm of a matrix by using an upper bound on the entries

Let $F:\mathbb{R}^3\setminus\{0\}\to\mathbb{R}^3$ be a function of class $C^1$ such that $$(x_1, x_2, x_3)\in \mathbb{R}^3\setminus\{0\}\mapsto F_i(x_1, x_2, x_3)\in\mathbb{R}, \quad \forall i\in\{1, ...
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2 votes
0 answers
24 views

Topological conjugacy for vector field failing on the level of flows

Given two vector fields $F_1,F_{2}: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with their flows $\phi_{1}, \phi_{2}:\mathbb{R} \times \mathbb{R}^{n}\rightarrow \mathbb{R}^n$ We say they are topological ...
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0 answers
14 views

Recursive sequence from piecewise-linear interpolation of $x\mapsto x^{p/q}$ for $p/q\in(0,1)$ only seems to converge to it if $p/q=1/2$

So, recently i got a bit (too) curious about approximating $x\mapsto x^{p/q}$ for arbitrary values of $p/q\in{]0,1[}$, starting with the following piecewise linear interpolation for all integer $n\ge1$...
0 votes
1 answer
78 views

About limit of an integral

Let $J=[0,1]$ and $x\in C[0,1]$, that is a continuous function on $[0,1]$. Can I say that $$\lim_{n\to \infty}\int_{\frac{1}{n^2}}^1\left\rvert \frac{1}{\sqrt{t}}-x(t)\right\lvert\;dt =\int_0^1\left\...
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15 votes
4 answers
352 views

Using an integral to generate rational approximations of $\pi$

Let $$f(r)=\int_0^1\frac{x^r(1-x)^r}{1+x^2}dx.$$ One surprising fact is that $f(4)=\frac{22}{7}-\pi.$ This got me thinking. Surely, it can't be a coincidence that a fairly accurate rational ...
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1 vote
1 answer
40 views

$\sigma$-finiteness of measures and separability of $L^p$ spaces

The fact that a measure $\mu$ is $\sigma$-finite determines or not the separability of $L^2(\mu)$? I proved that $L^2([0,1])$ endowed with the counting measure $m$ is not separable since it admits an ...
1 vote
1 answer
38 views

Proving that the non-empty finite subset of an ordered set is bounded

Let S be an ordered set. Let $A \subset S$ be a nonempty finite subset. Then prove that A is bounded. Furthermore, show that inf A exists and is in A and sup A exists and is in A. Hint : Use Induction....
1 vote
1 answer
39 views

Prove that a decreasing sequence of closed and bounded sets converges in the Hausdorff metric

Let $(X,d)$ a complete metric space. I need to prove that $(\mathcal{CB}(X),d_H)$ is complete. I see other posts (yes, it's a duplicate but no one answered this question) and the only thing I couldn't ...
-2 votes
1 answer
52 views

$\underset{n\rightarrow\infty}{\lim}\left(f\left(\left(n+1\right)^{2}\right)-f\left(n^{2}\right)\right)=0$ [closed]

Prove that for each function that maintains $0\le f'(x) \le \frac{1}{x^2}$ for $x>0$ the limit is $$\underset{n\rightarrow\infty}{\lim}\left(f\left(\left(n+1\right)^{2}\right)-f\left(n^{2}\right)\...
3 votes
2 answers
120 views

An asymptotic expansion for $u_{n+1} = u_n^2 + u_n$

I'm completely stuck finding an equivalent of the following recurring sequence. $\left\{\begin{matrix} u_0 > 0\\ u_{n+1} = u_n^2 + u_n \end{matrix}\right.$ The problem suggests to use the sequence ...
2 votes
1 answer
44 views

Convergence of the sequence $\sum \frac{n+5}{\sqrt{n^5 - 10n+10}}$

I am stuck on a question and would really appreciate if anyone can give some help. Determine whether the series: $$\sum_{n=1}^{\infty} \frac{n+5}{\sqrt{n^5 - 10n+10}}$$ converges or not. I am trying ...
1 vote
2 answers
36 views

Show that $f'(y) = 0$ when $f$ has a local maximum at $y$ [duplicate]

Suppose $f: [0,1] \to \mathbb{R}$ is a differentiable function. For some given $y \in (0,1)$, $\exists$ $\epsilon > 0$ such that $$f(x) \leq f(y) \ \forall \ x \in (y-\epsilon, y+\epsilon).$$ Show ...
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0 votes
0 answers
37 views

Finding a suitable form factor for given conditions

This is basically a physics problem but I will try my best to highlight the mathematics behind it. Suppose I have two functions: $$T(z,B)=\frac{\text{z}^3 e^{-3 A(\text{z})-B^2 \text{z}^2}}{4 \pi \...
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0 votes
0 answers
38 views

If $\lim (x_n)=0$ and $a>0$ then $\lim a^{x_n}=1.$

If $\lim (x_n)=0$ and $a>0$ then $\lim a^{x_n}=1.$ The solution given is as follows: We have $\lim a^{\frac 1n}=1$ and $\lim a^{-\frac 1n}=1.$ Let us choose $\epsilon>0.$ There exist natural ...
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-1 votes
0 answers
36 views

Techniques to solve limits of sequences

Often times in sequences and series I come across problems which cannot seem to yield with my limited manipulation ways. I get stuck,and since I do not have a lot of time to keep thinking about such ...
0 votes
0 answers
29 views

Let $\alpha$ be a cut and $r\in\Bbb Q^+.$ Then $\exists p\in \alpha$ and $q\in\alpha ^c$ such that $q$ is not the least element of $\alpha^c$ and

I was studying real analysis. I recently got to know the definition of a cut. The definition goes as follows: A set of $\alpha$ of rational numbers is said to be a cut if : (i) $\alpha\neq \emptyset, ...
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3 votes
2 answers
42 views

Show that this function is $C^1$ class.

Let $f:[-1,1]\to\mathbb R$ be $C^1$ class. Define $g:[-1,1]\to\mathbb R$ by $$g(x)=\begin{cases}f(0) &\mathrm{if}\ x=0 \\ \frac{1}{x}\int_0^x f(t)dt &\mathrm{otherwise} \end{cases}$$ Then, ...
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0 votes
1 answer
46 views

If $f: [0, \infty) \to [0,\infty)$ is concave, show that $f(x+y) \leq f(x) + f(y)$ $\forall \ x,y$ in the domain. [duplicate]

If $f: [0, \infty) \to [0,\infty)$ is concave, show that $f(x+y) \leq f(x) + f(y)$ $\forall \ x,y$ in the domain. We can assume $f$ to be strictly increasing on the interval $I = [0, x+y]$ since if $...
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0 votes
1 answer
20 views

Alternate standard definition of convergence of a sequence

Is this a standard definition of convergence of a sequence? $\{x_n\}$ converges if and only if for any given $\epsilon > 0$, $\exists \ N \in \mathbb{N}$ such that $|x_{n+1} - x_n| < \epsilon$ $...
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0 votes
1 answer
40 views

Equivalent definitions of Baire spaces

We say that a metric space $X$ is a Baire space if there is no open set $E$ such that $$E \subseteq \bigcup\limits_{n\geq 1} F_i,$$ in which each $F_i$ is a closed set with empty interior. Suppose ...
3 votes
0 answers
30 views

Riemann-Liouville Fractional Integral operator in $L^p(a,b)$ : finding the operator norm

I've recently come across the Riemann-Liouville operator in $L^p(a,b)$ (with $1\leq p \leq +\infty$), whose definition is : $V^\alpha > f(t)=\dfrac{1}{\Gamma(\alpha)}\int_a^t(t-x)^{\alpha-1}f(x)dx$...
-1 votes
0 answers
53 views

$ \lim_{x\to a}f(x)=L$ $\iff$ $ \lim_{x\to a}f_i(x)=L_i$

f : $\mathbb{R^n}\to\mathbb{R^m}$, L $\in\mathbb{R^m}$ I could prove that $ \lim_{x\to a}f(x)=L$ $\implies$ $ \lim_{x\to a}f_i(x)=L_i$ like this Assume $ \lim_{x\to a}f(x)=L$ Given $\epsilon>0\ \...
1 vote
2 answers
46 views

Proof that the set $(1, 2]$ is a open ball in $(0, 2]$ with euclidean metric

I want to prove that the set $M=(1, 2]$ is a open ball in $M=(0, 2]$ with euclidean metric, what I think is that I must prove that for a center $p$ and a radius $r>0$ $$B(p; r)=\left\lbrace x\in M:...
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1 vote
0 answers
32 views

Brezis's Proposition 4.21: did the author mean $f \star \rho_n$ rather than $\rho_n \star f$?

For $f \in L^1\left(\mathbb{R}^N\right)$ and let $g \in L^p\left(\mathbb{R}^N\right)$ with $1 \leq p \leq \infty$, we define $$ (f \star g)(x)=\int_{\mathbb{R}^N} f(x-y) g(y) d y. $$ A sequence of ...
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1 vote
0 answers
25 views

The function $\rho$ defined by $\rho (x) := \begin{cases} e^{1/ (|x|^2-1)} &\text{if } |x| <1 \\ 0 &\text{otherwise} \end{cases}$ is smooth

We define $\rho:\mathbb R^n \to \mathbb R$ by $$ \rho (x) := \begin{cases} e^{1/ (|x|^2-1)} &\text{if } |x| <1, \\ 0 &\text{otherwise} \end{cases} $$ It's is mentioned at page $108$ of ...
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1 vote
1 answer
69 views

Given an uncountable nowhere dense set $A$ of measure zero, is it true that $A+A$ has a positive measure?

Suppose $(\mathbb{R}, \Omega, \mu)$ is Lebesgue measure space. Given an uncountable nowhere dense set $A \subset \mathbb{R}$ such that $\mu(A) = 0$, is it true that $A+A = \{a + b \in \mathbb{R}: a,b ...
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0 votes
0 answers
35 views

Prove that $\lim_{n \to \infty} m^{1/q(n)} = 1$

It is required to prove that $\lim_{n \to \infty} m^{1/q(n)} = 1$, where $m$ is a natural such that $m \ge 1$ and $\{q(n) \in [1, \infty): n \in \mathbb{N}\}$ diverges to $\infty$. It has already been ...
0 votes
1 answer
44 views

If $ f $ is increasing on $ (a, b) $ and for some $ x_0 \in (a, b) $, $ \lim_{x \to x} f(x) = L $, then $ f $ is continuous at $ x_0 $

Let $ f $ be a monotonically increasing function on the interval $ (a, b) $ and let $ x_0 \in (a,b) $ such that $ \displaystyle \lim_{x \to x_0} f(x) = L \in \mathbb{R} $. I need to prove that $ f \ $ ...
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0 votes
1 answer
39 views

Proving that the two characterizations of integrability are equivalent

I am self-learning Real Analysis from the text Understanding Analysis, by Stephen Abbott. I am trying to prove that two characterizations of the Riemann integral are equivalent. The stated definition (...
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1 vote
4 answers
120 views

Solving the integral: $\int_{-\infty}^{+\infty} x^n e^{-ax^2+bx} \, dx$... Moments of anormal distribuition?

I'm trying to find a closed formula for the integral in $(eq. 1)$, but i wasn't able to do anything. On the other hand, while searching on the web i found a closed formula for the cases $(n=1, \, n=2)$...
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2 votes
0 answers
32 views

Metric on bounded sequences that induces pointwise convergence

I got the following question in my functional analysis course. let B($\mathbb{N}$) be the linear space of bounded functions $f:\mathbb{N}\rightarrow\mathbb{R}$ (i.e bounded sequences). Find a metric d ...
1 vote
0 answers
20 views

Counterexample of Sobolev Embedding Theorem $W^{1,n}(\mathbb{R}^n) \nsubseteq L^\infty (\mathbb{R}^n)$ [duplicate]

I am looking for a counterexample of Sobolev Embedding Theorem $W^{1,n}(\mathbb{R}^n) \nsubseteq L^\infty (\mathbb{R}^n)$. I feel that somehow the log(log()) function could provide an example but I am ...
4 votes
3 answers
91 views

How can I compute the $n$-th derivative of $e^{a e^{-x}}$

I'm trying to compute the $n$-th derivative of the function $e^{a e^{-x}}$, but I'm stuck. I found that I could use Faa di Bruno's formula but maybe I'm worsening the problem to the computation of the ...
0 votes
0 answers
24 views

How do I show the differentiation under integral sign when Leibnitz rule does not work?

In proving Taylor's theorem with remainder (in Tu's Manifold version) I'm stuck at showing that $\int^1_0 \frac{\partial f}{\partial x^i} (p+t(x-p))dt$ is smooth. Here $f$ is smooth and it is defined ...
-1 votes
0 answers
85 views

Most cited areas in Pure Mathematics [closed]

which areas of Pure Mathematics are the most cited? I cannot find any concrete data for this; all I can find are the most cited mathematicians, not the most cited fields. It appears as though Number ...
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1 vote
0 answers
26 views

Norm of Hermite polynomials in a non-standard Gaussian

The Hermite polynomials I am using satisfy the recurrence $H_k'(x) = kH_{k-1}(x)$ and they satisfy the property if $d\mu_{1/2} = (2\pi)^{-1/2}e^{-x^2/2}$ is the standard Gaussian then $||H_k||_{L^2(\...