Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, the least upper bound property; and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus. This tag can also be used for more advanced topics, like measure theory.

Filter by
Sorted by
Tagged with
0 votes
0 answers
7 views

Is this an abuse of notation? (Limits)

This exercise is from my textbook: Let $I\subseteq \mathbb{R}$ be an open interval, let $f : I\rightarrow \mathbb{R}$ be differentiable on $I$, and suppose $f''(a)$ exists at $a\in I$. Show that: $$f''...
user avatar
  • 43
0 votes
0 answers
7 views

Proof of Lagrange multipliers theorem

I am currently studying Folland's advanced calculus. But it doesn't have a rigorous proof of this theorem, so I try to write one myself. Let $f$ be a differentiable mapping of $\mathbb{R}^n$ into $\...
user avatar
0 votes
0 answers
11 views

Measurability of coupling of two probability distributions

Let the sample space be $\mathbb{R}^d$ and $P$ be a probability transition kernel on $\mathbb{R}^d$. Let $x,y\in \mathbb{R}^d$ and thus $P(x,\cdot),P(y,\cdot)$ are probability measures and for each $x,...
user avatar
  • 3,126
1 vote
0 answers
10 views

Observation about some points

I want to construct a point set where each point has the same (w.l.o.g., unit) distance to all other points in the $L_1$ metric. Example: The points $\left(\frac{1}{2},0\right)$, $\left(-\frac{1}{2},0\...
user avatar
  • 11
-3 votes
1 answer
33 views

How can I use the first Fundamental theorem of calculus for finding f that satisfy: [closed]

$(f(x))^2=\displaystyle\int _0^x f(t)\frac {\sin (t)}{3+\cos (t)}dt$
user avatar
2 votes
0 answers
16 views

Bounding diameter of the arc of a closed curve

I was reading chapter 4 of Colding and Minicozzi's A Course in Minimal surfaces and I came across a statement in the proof of Lemma 4.14: Suppose $\Gamma\subset\mathbb{R}^3$ is a simple closed curve ...
user avatar
  • 485
0 votes
0 answers
19 views

Proving that for any irrational number there is a irrational numbers and rational numbers sequence converging to it [duplicate]

I am struggling with how to prove that for any irrational number there are sequences of rational numbers that converge to it but also a sequence of irrational numbers converging to it. That is a ...
user avatar
0 votes
0 answers
17 views

Any hints on how to prove that the given equation has solutions only when the RHS is positive?

I have this equation for $x>0$, $\beta>0$ and $\alpha$ (all reals) $$ \frac{\alpha \cos (\beta x)}{x}+\frac{\alpha \cos (2 \beta x)}{x}-\alpha \beta \cot \left(\frac{\beta x}{2}\right)+4 \...
user avatar
  • 73
0 votes
1 answer
20 views

Can a function mapping any set to its lower upper bound be bijective?

I wonder whether a function mapping any bounded set of reals A to a number that is set's least upper bound be both surjective and injective? Is it the same with function mapping to set's greatest ...
user avatar
0 votes
0 answers
14 views

How to create a continuous spiral function?

I'm trying to make a continuous function $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ that looks like a spiral. So far I managed to come up with this: $$ f(x,y) = \begin{cases} \sqrt{x^2 + y^...
user avatar
4 votes
1 answer
27 views

Sequence of random variables converging to zero arbitrarily slowly

The following question occurred to me: Suppose $X_n$ is a sequence of positive random variables satisfying for all $\delta>0$, $P(X_n < \delta) \to 1$. Is it true that there must exist a ...
user avatar
4 votes
1 answer
22 views

Rank 1 operator on an infinite dimensional vector space number of eigenvalues.

Does a rank 1 bounded operator from $\mathscr{K}:L^2([0,1])\to L^2([0,1])$ have at most 1 non-zero eigenvalue? The reason this is not obvious to me is that $L^2([0,1])$ is infinite dimensional. In ...
user avatar
  • 1,262
1 vote
0 answers
22 views

Log-normal distribution: Why is this a density function?

I want to prove that $$f(x) = \frac{1}{\sigma x \sqrt{2\pi}} \exp(-\frac{\ln(x)^2}{2\sigma^2})$$ for $x>0$ and $f(x)=0$ for $x\leq 0$ is a density function, with $\sigma > 0$. So what I have to ...
user avatar
  • 449
0 votes
0 answers
4 views

The Clarkson's inequalities in the extended real case deduced from the complex case

In Adams' text Sobolev Space there is an exhaustive treatment of Clarkson's inequalities in the case in which the functions of $L^p$, where $p\in(1,\infty)$ are complex-valued. Do the same proofs hold ...
user avatar
  • 159
-1 votes
0 answers
13 views

$cl(A) \cap cl(B) = \emptyset$, then proof that $bd(A \cup B) = bd(A) \cup bd(B).$

Let $A$ and $B$ are two subsets of a metric space $(X,d)$. Suppose that $cl(A) \cap cl(B) = \emptyset$, then proof that $bd(A \cup B) = bd(A) \cup bd(B).$ Where $cl(A)$ is the closure of the set $A$ ...
user avatar
0 votes
0 answers
11 views

Trouble understanding a criteria for integrability

Let $f : [0, 1] → \mathbb{R}$ be bounded and $f ≥ 0$. Prove that if the set $\{x\in[0,1]|f(x)\geq \lambda \}$ is finite for every $λ > 0$ then $f$ is integrable. I'm having trouble understanding ...
user avatar
  • 490
1 vote
0 answers
23 views

An exercise on counting fixed points from Milnor

I am working on the following exercise (Problem 6-b, pg. 6-22) from Milnor's Dynamical Systems notes: Problem 6-b Let $f$ be any self map such that each iterate has only finitely many fixed points. ...
user avatar
  • 11
0 votes
0 answers
20 views

quadratic function cutoff analysis,some ideas?

I have a parabola of the form $f(x)=ax^2+bx+c$ What condition must the coefficients a,b,c, meet, so that the parabola intersects the x-axis at two points? a) in the negative part of x b) on the ...
user avatar
3 votes
1 answer
31 views

If $\int_X f d\mu = M$, then there $x_0 \in X$ such that $f(x_0) \ge M$

Let $X$ be a metric space and $\mu$ a Borel probability measure on $X$. Let $f \in L_1(X, \mu, \mathbb R)$ such that $\int f d\mu = M$. Can we show that there is $x_0 \in X$ such that $f(x_0) \ge M$? ...
user avatar
  • 9,968
0 votes
1 answer
41 views

Check Understanding of $\varepsilon$-$\delta$ Limit Proof

Say, we want to prove that $\lim_{x \to a} x^2 = a^2\;$ Assuming $a>0$ here. Here’s how I would think the $\varepsilon$-$\delta$ proof way. Please give feedback on thinking. $$\forall \varepsilon&...
user avatar
0 votes
1 answer
40 views

Does this sequence converge uniformly?

We consider some sequence $(f^{(k)})_{k\in \mathbb{N}}\subset L^p(\mathbb{R}^3)$ such that $f^{(k)}\to f$ in $L^p(\mathbb{R}^3)$. Further assume there is some $R>0$ so that $\text{supp}(f^{(k)}),\...
user avatar
  • 552
-4 votes
0 answers
26 views

Let ${x_n}$ be a sequence such that $x_{n}^{3}$ converges to a nonzero number. Then the sequence $ \sin(x_{n}) $ converges. True or false? [closed]

Justify true or false. If it is true prove it other wise give counter example or proper reason.
user avatar
0 votes
1 answer
37 views

If $f$ non-negative and bounded and $\int_{\mathbb{R}}f d \lambda < \infty \Rightarrow \int_{\mathbb{R}}f^{2} d \lambda < \infty$

I am trying to show if $f$ is a non-negative function that is bounded and $\int_{\mathbb{R}}f d \lambda < \infty \Rightarrow \int_{\mathbb{R}} f^{2} d \lambda < \infty$ Where d$\lambda$ denotes ...
user avatar
2 votes
1 answer
21 views

Comparison of $L^p$ and $L^q$ norms to establish the inclusion between corresponding spaces

We can deduce that; for any $x \in \ell^p,$ the space of $p$-summable real sequences ($p \geq 1$), $$\lVert x \lVert_q \leq \lVert x \lVert_p,~p \leq q < \infty,$$ by just letting $e=\frac{x}{\...
user avatar
  • 350
1 vote
1 answer
33 views

Convergence of a series constructed from the elements of $\mathbb{Z}^k$

Let $\mathbb{Z}$ be the set of integers. For $z = (z_1, \ldots, z_k) \in \mathbb{Z}^k$, $k \in \mathbb{N}$, let $|z|_1 := |z_1| + \ldots + |z_k|$ denote the $l_1$-norm of $z$. Is it true that $$ \...
user avatar
  • 93
0 votes
0 answers
28 views

$\lim_{x \to 0+}{{x^\epsilon}\ln x} = \lim_{x \to \infty}{\ln x/{x^\epsilon}} =0$ for every $\epsilon>0$.

I want to show that $\lim_{x \to 0+}{{x^\epsilon}\ln x} = \lim_{x \to \infty}{\ln x/{x^\epsilon}} =0$ for every $\epsilon>0$. I know it can be easily proved when using L'Hôpital's rule but I'm not ...
user avatar
0 votes
0 answers
10 views

Density of the restriction of an absolutely continuous measure

Suppose we have a probability space $(X,\mathcal{M},\mu)$ and an absolutely continuous probability measure $\nu$ with respect $\mu$ (in symbols $\nu\ll\mu$ with density $\rho$). If I consider the ...
user avatar
0 votes
0 answers
17 views

Inequality $\theta\|{-\Delta u}\|_{L^2(U)}^2 \leq (Lu,-\Delta u),$ for elliptic operator

Let $U$ be the bounded smooth open subset of $\Bbb{R}^n$, with $u \in H^2 \cap H^1_0$. Let $L = \sum_{ij} (a_{ij}(x) u_{x^i})_{x^j} + \sum_k b_k(x) u_{x^k} + c(x) u$ be a general linear differential ...
user avatar
  • 2,968
1 vote
1 answer
48 views

Standard result on limits? But what is it?

Book's Question Suppose $g : \mathbb R\to \mathbb R$ is continuous at $0$, and let $h : (0, ∞) → R$ be defined by $h(x) = g(1/x)$. Prove that if $ h(n) > 0 $ for all $n \in \mathbb{N}$ then $g(0) ≥ ...
user avatar
1 vote
0 answers
24 views

dependence of multivariable functions

(I met this question in proving all the integrating factors $\mu_1$ of $P(x,y)dx+Q(x,y)dy=0$ has the form $\mu_1(x,y)=\mu(x,y) g(\Phi)$,where $\mu Pdx+\mu Qdy=d\Phi,\mu$ is also an integrating factor)....
user avatar
  • 381
-3 votes
0 answers
25 views

Does this sum containing a zero sequence converge to 0 [closed]

I am trying to understand a proof, where we have $(c_i)_{(i\geq 0)}$ a (real) sequence such that $\lim_{i \to \infty}c_i = 0$. I want to show that $$ \lim_{n \to \infty} \frac{1}{n}\sum_{i=0}^nc_i = 0....
user avatar
  • 17
1 vote
1 answer
20 views

Is there any $K_t^{\epsilon}\to \mathbf{1}_{[0,t]}$ such that $\frac{d}{dt}\|K_t^{\epsilon}\|^2\to 1$?

I am looking for conditions or at least one example such that the following holds true. Let $K_t^{\epsilon}:[0,T]\to\mathbb R$ be an smooth function such that $K_t^{\epsilon}\to \mathbf{1}_{[0,t]}$ in ...
user avatar
  • 3,119
0 votes
0 answers
18 views

The definition of connected is stated as the negation of disconnected, but a little care with the logical negation, the results in a characterization. [closed]

A set E is totally disconnected if, given any two distinct points x, y ∈ E, there exist separated sets A and B with x ∈ A, y ∈ B, and E = A∪B. (a) Show that Q is totally disconnected. (b) Is the set ...
user avatar
1 vote
1 answer
40 views

Understanding a certain solution , if $(a_n)$ is a strictly decreasing sequence then $(a_n)$ is positive

Let $(a_n)$ be a zero sequence ,if $(a_n)$ is a strictly decreasing sequence then $(a_n)$ is positive This is the solution in the textbook : Given $(a_n)$ is a strictly decreasing sequence ( $a_n &...
user avatar
  • 695
0 votes
0 answers
21 views

Showing $\lVert \Delta u \rVert_{L^2} + \lVert u \rVert_{L^2}$ is equivalent to $\|u\|_{H^2}$ norm for $H^2$ space

This question has been asked here showing $\lVert \Delta u \rVert_{L^2} + \lVert u \rVert_{L^2}$ is equivalent to $\|u\|_{H^2}$ norm for $H^2$ space However I came across some problem when following ...
user avatar
  • 2,968
0 votes
1 answer
27 views

Elliptic regularity for Poisson equation

Let $U \subset \Bbb{R}^n$ be an open bounded smooth domain, let $u \in H^1_0(U)$ satisfy the Poisson equation: $$-\Delta u = f \tag{*}$$ with $f \in L^{2}(U)$, by the classical elliptic regularity ...
user avatar
  • 2,968
0 votes
0 answers
39 views

Solution check: $e^{\cos x}=\sum_{n=0}^{\infty}{a_nx^n}$, determine $a_n$ for $n\leq 5$, does this solution make sense?

On my caculus book, there is a question: Suppose the Taylor expansion of $e^{\cos x}$ is $e^{\cos x}=\sum_{n=0}^{\infty}{a_nx^n}$, determine the value of $a_n$ for $n\leq5$. Here is the solution the ...
user avatar
  • 101
3 votes
0 answers
27 views

Are these equations "properly" defined differential equations?

Are these equations properly defined differential equations? Modifications were made to a deleted question to re-focus it Intro Recently, in this answer I figure out that the following autonomous ...
user avatar
  • 839
0 votes
1 answer
74 views

Prove if this inequality holds true

Let $N\in\mathbb{N}, N\ge 2$ and $p, q\in\mathbb{R}, 1<q<p<N$ and also $q>\frac{N(p-1)}{(N-1)}$. Under these assumptions, it is true that $$\frac{N-p}{p}- \frac{N(p-1)(p-q)}{p^2}>0?$$ I ...
user avatar
  • 795
0 votes
0 answers
13 views

Convergence in probability if and only if every subsequence has further subsequence convergence almost everywhere

Series of random variables $\{X_n\}$ converges in probability to $X$ if and only if its every subsequence $\{X_{n_k}\}$ contains a further subsequence that converges almost everywhere to $X$. Here is ...
user avatar
0 votes
0 answers
20 views

Investigate convergence, boundedness and if applicable the limit of the recursive sequence $(x_n)$ $\subset$ $\mathbb R$.

Investigate convergence, boundedness and if applicable the limit of the recursive sequence $(x_n)$ $\subset$ $\mathbb R$. Given are, $x_0$ := 0 ; $x_{n+1}:= $$\frac{1}{2}(1-x_n^2)$ For showing ...
user avatar
  • 1
1 vote
1 answer
52 views

How to apply intermediate value theorem to this quantity?

Let $N\in\mathbb{N}$ with $N\ge 2$ and $s, p\in\mathbb{R}$ with $s, p>1, p^*=Np/(N-p)$. Let $(x_n)_n$ be a sequence such that $x_n\to x_0$. Consider the quantity $$\frac{1}{(s+1)^{p-1}}(x_n^{p(s+1)}...
user avatar
  • 2,353
-1 votes
1 answer
68 views

Prove that $\lim_{n \to \infty} \frac{\log(n+1)}{\log(n)} = 1$ [closed]

I have tried using a property of logarithm, with $$\frac{\log(n+1)}{\log(n)} = \log_{n}{(n+1)}.$$ However, I don't know how to proceed or if I'm really doing it the right way. Please, does someone ...
user avatar
  • 1
1 vote
0 answers
23 views

Generalization of this inequality?

Here is the statement which has been given : Let $A \in \mathbb{R}^*_+$, $f \in \mathcal{C}^1([0,A],[0,f(A)])$ a strictly increasing function with $f(0)=0$. Then for all $x \in [0,A]$ and $y\in [0,f(...
user avatar
  • 2,937
4 votes
0 answers
49 views

Is there any hope to prove that the given function is always negative if the given condition holds?

I have this function for $x>0$ and $\alpha,\beta\in\mathbb{R}$ and $\alpha>0$ $$f(x)=-4 \alpha x^2+4 \alpha x^2 \cos (\alpha x)+\alpha \beta x \sin (\alpha x)-\beta \cos (\alpha x)-\beta ...
user avatar
  • 73
-1 votes
0 answers
15 views

Find zeros of a parameterized function

Let $a \geq 0$ and $b \geq 0$. Define the function $f(x) = e^x+e^a (x-b+1)$. I have difficulties characterizing the zeros of $f$ in terms of $a$ and $b$. The online solvers that I've tried so far also ...
user avatar
  • 121
0 votes
0 answers
23 views

Difference between two similar definitions of integrability

Let $f ∈ TF([a, b])$ (set of all step functions) be a bounded,positive function and let $T_U$ be the set of step functions with $u ≤ f$, $\forall u ∈ T_U$ and $T_O$ be the set of step functions with $...
user avatar
  • 490
3 votes
2 answers
90 views

Is it possible to find the maximum of $r\ln\left(\ln\left(1+\frac{1}{r}\right)\right)$ for $r\in(0,1)$?

I was working on Chapter 5, Problem 14 of Evan's Partial Differential Equations 2nd ed and to prove integrability of $\ln \left (\ln \left (1+\dfrac{1}{|x|}\right )\right )$ on the unit $n$-ball, ...
user avatar
  • 51
0 votes
0 answers
13 views

Approximating minimizers with $\epsilon$-nets

Let $K$ be a metric space. Fix $\epsilon > 0$. We say that $E \subset K$ is an $\epsilon$-net of $K$ if for each $x \in K$ there exists $e \in E$ such that $d(x,e) < \epsilon $. Consider a ...
user avatar
  • 1,666
1 vote
1 answer
40 views

Finding functions $f, g, h : \mathbb R_{> 0} \to \mathbb R_{> 0} $

Undergraduates at my university showed me this problem, which I found intriguing and now want to see the solution of: Find all functions $f, g, h : \mathbb R_{> 0} \to \mathbb R_{> 0} $ such ...
user avatar

1
2 3 4 5
2641