For questions concerning the definition and properties of limit superior and limit inferior.

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Prove or disprove: $\sup\left \{ x\in\mathbb{R}\mid x^2-5x+6\leq0 \right \}=3$

No homework:http://www2.mathematik.hu-berlin.de/~gaggle/S15/MATHINFO/UEBUNG/nachholklausur.pdf Prove or disprove: $\sup\left \{ x\in\mathbb{R}\mid x^2-5x+6\leq0 \right\} =3$ I would say the ...
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0answers
31 views

Can you explain for me why author could put $\alpha_{n,k}$ as above?

I have a theorem: Let $\{a_n\}$ be a sequence of complex numbers and let $\{A_n\}$ be a non-decreasing sequence of positive numbers, tending to infinity. The function $N(t) = \#\{n ≥ 1 : A_n/|a_n| \...
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1answer
18 views

Relation between Supremum and limit superior

If $\sup\limits_{n\ge 1} a_n<\infty$ then obtains that $\limsup\limits_{n\to\infty} a_n<\infty.$ Can you explain that?
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2answers
30 views

inf, sup, max, min for $\bigcap_{n \in \mathbb N}\left[-\frac{1}{3^n},4+\frac{1}{2n}\right)$

For the following set do I have the inf, sup, min, max correct? $\bigcap_{n \in \mathbb N}\left[-\frac{1}{3^n},4+\frac{1}{2n}\right)$ $\text{inf}S=0$ $\text{sup}S= 4$ $\text{min}S=0$ $\text{...
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0answers
28 views

How to draw conclusion about Limits

Let $f$ and $g$ be two $\mathbb{R}\to\mathbb{R}$ functions such that $$ \limsup_{t\to\infty}\frac{1}{t}\log\left(\left|f(t)\right|\right)>\limsup_{t\to\infty}\frac{1}{t}\log\left(\left|g(t)\right|\...
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1answer
154 views

A surprising inequality about a $\limsup$ for any sequence of positive numbers

I think I remember this from either Apostol's or Rudin's principles of Analysis. Claim: Let $\{x_n\}_{n=1}^{\infty}$ be a sequence of positive real numbers. Then $$ \limsup \frac{x_1+x_2+...+x_n+x_{n+...
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1answer
27 views

Proofs involving limsup and liminf

I've been working with proofs involving $\limsup$ and $\liminf$, and I'm a bit confused regarding their general methodology. More specifically, I'm unsure about whether my approach to the following ...
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1answer
22 views

Radius of convergence of integral series; problem with limsup

Let $\sum c_k x^k$ be a power series with radius of convergence $R$. Then the integral series $$\sum_{k=0}^\infty \frac{c_k}{k + 1}x^{k+1}$$ also has radius of convergence $R$. I'm reading Real ...
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2answers
37 views

On a superior limit involving the multiplication formula for the Gamma function and the divisors $d\mid n$ of a positive integer

I did the specialization for the $m's$ in the multiplication formula for the Gamma function, see the identity (4) in page 250 of Apostol, Introduction to Analytic Number Theory Springer (1976) as the ...
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1answer
76 views

Prove that $u$ is upper semicontinuous on $\Delta(0,\rho)$.

Let $u:\Delta(0,\rho)\rightarrow \mathbb{R}$ be a function such that $u(x+iy)$ is convex in $x$ for each fixed $y$, and convex in $y$ for each fixed $x$. Prove that $u$ is subharmonic on $\Delta(0,\...
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1answer
35 views

Upper limits for $s_n$ $\leq$ $t_n$ [duplicate]

Let $(s_n)$ and $(t_n)$ be two sequences of real numbers. Suppose there exists $N_0$ such that for all $n>N_0$, $s_n \leq t_n$. Two Questions: Suppose that $\lim s_n$ and $\lim t_n$ both exist. ...
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3answers
32 views

Supremum, max, infimum, min of a set

$S=\bigcap_{n \in \mathbb N}\left(3, \frac{7n+1}{n}\right)$ I still haven't full grasped how to solve these nested interval intersections problem as I have issues visualizing what is happening as $n$...
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1answer
61 views

Is this an open or closed set?

$S=\{5+\frac{(-1)^n}{n} \; : \; n \in \mathbb N\}$ According to my calculations this set has a lower bound of $4$ and an upperbound of $5$; however, since $4$ is reachable by the set it is a minimum ...
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1answer
19 views

How can I find the subsequential limit, limit sup, and limit inf of $s_n=n\tan\frac{n\pi}{3}$

$s_n=n\tan\frac{n\pi}{3}$ How can this sequence be decomposed to the the set of subsequences so that I can find the limit sup, and limit inf? I suppose I could just take $n$ and then $\tan\frac{n\pi}...
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2answers
26 views

Subsequential limit of $S_n=\left(-\frac{\sqrt{2}}{4}\right)^n$

I'm having trouble identifying the subsequences, or patterns of the sequence. I recognize the problem can be rewritten: $S_n=\left(-\frac{\sqrt{2}}{4}\right)^n=-\frac{\sqrt{2}^n}{4^n}$ And if I ...
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0answers
24 views

Show that $f$ defined on the interval $(a,b)$ is not differentiable for every point in $E$ with $m(E)=0$

Let $E$ have measure zero contained in the open interval $(a,b)$. In a previous problem I showed that there is a countable collection of open intervals, $\{(c_k,d_k)\}_k$, contained in $(a,b)$ for ...
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1answer
31 views

$\limsup$ sequence independent $\mathcal{N}(0,\sigma^2)$

In my lecture notes there is the following application of Borel-Cantelli's 2nd lemma: Let $(X_n)_{n\geq 1}$ be a sequence of independent $\mathcal{N}(0,\sigma^2)$-distributed random variables, with $\...
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0answers
22 views

Probability that there exists $M>0$ such that two processes $\{X_t\}$ and $\{Y_t\}$ are smaller than $M$ at the same time, for infinitely many $t$.

Suppose I have two iid sequences of random variables $\{X_t\}_{t\in\mathbb{N}}$ and $\{Y_t\}_{t\in\mathbb{N}}$, both absolutely continuous with full support. I know that with probability one, for any $...
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1answer
41 views

$\limsup_{k\rightarrow\infty}X_k/k$ for identically distributed random variables $X_k$.

Let $\{X_k\}_{k\in\mathbb{N}}$ be a sequence of identically distributed and not independent random variables on the natural numbers. I am interested in conditions that would ensure that almost surely $...
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1answer
34 views

Liminf, Limsup inequalities in Cesàro's Lemma proof

Probability with Martingales: I tried writing out the details of the proof avoiding $\ge$ if I felt it was unnecessary. Please tell me if I got any steps wrong. $$\liminf \frac{1}{b_n} \...
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0answers
37 views

Cesàro's Lemma - precise definition of limit, indices

Probability with Martingales: I'm a little confused about the precise langauge. I guess we have that $$\forall M > 0, \exists N_b > 0 \ \text{s.t.} \ b_n > M \ \text{whenever} \...
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2answers
82 views

Strong Law of Large Numbers - Converse

Probability with Martingales: I want to try to show the last one $$\left[\limsup \frac{|S_n|}{n}\right] = \infty \ \text{a.s.}$$ which is equivalent to $$\forall k \in \mathbb N$$ $$\left[\...
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1answer
34 views

Prove $\lim \sup f (x_n) = f(\lim \sup (x_n)) $ and same for $\inf$

Prove: Let $A \subset \mathbb{R}$ compact, $f: A \rightarrow \mathbb{R}$ continuous, increasing monotone and $(x_n) \subset A$. Consider. Show that $\lim \sup f (x_n) = f(\lim \sup (x_n))$ and $\lim \...
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1answer
42 views

$\inf_{x\in A}{\limsup_nd(x_n, x)} = \limsup_n[\inf_{x\in A}d(x_n, x)] $ for compact subset $A$.

let $ (X, d) $ be a complete metric space, $ A\subset X $ be compact and take a sequence $ (x_n) \subset X $\ $ A $ as a bounded sequence. Since infimum is independent from n , does the following ...
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1answer
20 views

What are $\lim \inf{\mathbb{Z}^+}$ and $\lim \sup{\mathbb{Z}^+}$ if $A$ is finite?

Definition. Let $A$ be a nonempty subset of $\mathbb{R}$. $x \in \mathbb{R}$ is called an almost upper bound of $A$ if there are only finitely many $y \in A$ for which $y \geq x$. Similarly we define ...
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30 views

Examples 3.35 (a) and (b) in Baby Rudin: Limit Superior and limit inferior of a couple of sequences

This question is related to Examples 3.35 (a) and (b) in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition, p. 67. Let us consider the series $$ \frac 1 2 + \frac 1 3 + \...
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63 views

unbounded solution, lim inf of trace,

Show that if $\lim \inf_{t\rightarrow \infty} \int_{t_0}^t \operatorname{tr}\left(A(s)\right)ds= \infty $ then the linear first-order system $x'(t)=A(t)x(t)$ where $A \in C\left(I, \mathbb{R}^{n\times ...
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1answer
24 views

When does $\max \lim{a, |b|} \leq a + \max \lim {|b|}$?

Let $\alpha_\delta >0$ be a quantity that depends only on $\delta$ and let $I_\varepsilon$ be defined as follows: $$I_\varepsilon = \alpha_\delta + \beta_{\varepsilon, \delta}$$ where $\beta_{\...
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0answers
31 views

Therem 3.17 in Baby Rudin: The Analogous Result

Here's Theorem 3.17 in the book Principles of Mathematical Analysis by Walter Rudin, third edition. Let $\{s_n\}$ be a sequence of real numbers. Let $E$ denote the set of all the subsequential ...
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1answer
30 views

$\limsup_{x\to 0}$ and $\liminf_{x\to 0}$

I want to find $\lim_{x→0}\sqrt{1+x+x²}=1$ and want to show that $\sqrt{1+x+x²}-1/(\sqrt{1+x}-\sqrt{1-x})$ tends to a limit as $x\to 0$ So in the first case I want to show that the $\limsup_{x\to 0}$ ...
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28 views

Help solve a Limit Question?

See this . What he's meant that "in particular"? where the $|g(x)|<|M|+1$ formula from? How deduced? What is the meaning of it?
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1answer
46 views

Prove $X_{\infty} < \infty$

From Williams' Probability with Martingales: How exactly do we prove $X_{\infty} < \infty$ a.s.? $$E[|X_{\infty}|] = E[|\lim X_n|] = E[|\liminf X_n|] = E[\liminf |X_n|]$$ $$ \le \liminf E[...
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87 views

Explicit Integrals and LimInf/LimSup

(a) Show that $f(t):=\int_0^\infty e^{-tx}\frac{sin \space x}{x}dx$ exists for $t>0$ and defines a differentiable function $f$. Calculate $f'(t)$ for $t>0$ and evaluate it explicitly. (b) Prove ...
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0answers
28 views

limsup and liminf of a function explenation

I have a little problem at understanding limsup and liminf of a function at a point y or infinity..I would like to understand the definitions and the theorems formally and intuitively..Can someone ...
2
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0answers
56 views

Theorem 3.37 in Baby Rudin: $\lim\inf\frac{c_{n+1}}{c_n}\leq\lim\inf\sqrt[n]{c_n}\leq\lim\sup\sqrt[n]{c_n}\leq \lim\sup\frac{c_{n+1}}{c_n}$

Here's Theorem 3.37 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: For any sequence $\{c_n\}$ of positive numbers, $$\lim_{n\to\infty} \inf \frac{c_{n+1}}{c_n} \...
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1answer
28 views

Help solving a Limit quesiton [closed]

enter image description herewhy can't just use 2x^3? enter image description herePrevious page
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$\lim _ {n\rightarrow \infty }\inf \{f_n \leq t \}=\{\lim _ {n\rightarrow \infty }\sup f_n \leq t \}$ Proof and Intuition

For a start how can I read $\lim _ {n\rightarrow \infty }\inf \{f_n \leq t \}$ ? I only seen $\lim _ {n\rightarrow \infty }\inf $ for sets and just don't understand the meaning of $\lim \inf$ for an ...
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1answer
33 views

Proof of the Strenghtened Limit Comparison Test

I'm studying on my own using Bonar and Khoury's Real Infinite Series. I understand the proof of the "regular" Limit Comparison Test( a link to google books, page 23 ) but the book doesn't provide a ...
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2answers
55 views

Theorem 3.19 in Baby Rudin: The upper and lower limits of a majorised sequence cannot exceed those of the majorising one

Here is Theorem 3.19 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. Let $\{s_n \}$ and $\{t_n \}$ be sequences of real numbers. If $s_n \leq t_n$ for $n \geq N$, ...
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0answers
44 views

Theorem 3.17 in Baby Rudin: Infinite Limits and Upper and Lower Limits of Real Sequences

Here're Definitions 3.15 and 3.16 and Theorem 3.17 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. Definition 3.15: Let $\{s_n \}$ be a sequence of real numbers ...
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3answers
34 views

Proof about infinite limsups and subsequences

I am working on a final exam study guide, and came across this question: Suppose limsup$(a_n)$ = $\infty$. Prove: There must exist a sub-sequence ${a_n}_k$ such that ${a_n}_k \to \infty$. My initial ...
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1answer
15 views

Equivalence bounded limit superior

Suppose that $(x_n)_{n=1}^\infty$ is a real sequence such that $\limsup_n x_n$ exists. I wish to show that $\limsup_n x_n\le\beta \iff \forall\varepsilon>0\ \ \exists N\ \ \forall n\ge N, x_n &...
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2answers
25 views

Let $A\subset \mathbb{R}$ such that $l=\text{inf }(A)$ exists. Prove that $\forall \epsilon >0 $ there is $a\in A$ in the interval $[l,l+\epsilon)$

I need to prove the following: Let $A\subset \mathbb{R}$ such that $l=\text{inf}(A)$ exists. Prove that $\forall \epsilon >0 $ there is $a\in A$ in the interval $[l,l+\epsilon)$ That's what I did:...
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1answer
37 views

Fatou Lemma: Why is $\lim\inf f_n = 0$ where $f_n = \chi_{[n,n+1]}$

In this wildly popular post, there is a claim: I like to remember this by example; specifically let $f_n = \chi_{[n,n+1]}$. Then $\lim \inf f_n = 0$, and $\lim \inf \int f_n = 1$. So $f_n = \...
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1answer
25 views

Convergence of a series implies convergence of another series

Let $a_1,a_2,\cdots$ be a sequence of real numbers with $a_i\geq 0$. If $\sum_{n=1}^{\infty}\frac{1}{1+a_n}<\infty$ then show that $\sum_{n=1}^{\infty}\frac{1}{1+x_na_n}<\infty$ for each real ...
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0answers
45 views

Infimum & Supremum in $\epsilon-\delta$ Proofs

When we are introduced to $\epsilon-\delta$ proofs in a usual first-year introductory course to Calculus, we usually always tend to do one of two things when we attempt to prove limits using the $\...
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1answer
12 views

I do not understand proving some limit superior problem.

Prob. Show that $~~~\displaystyle\limsup_{k\to\infty} (a_k+b_k) \le \limsup_{k\to\infty} a_k + \limsup_{k\to\infty} b_k$. Let $A_j=\displaystyle\sup_{k\ge j}a_k,~~~B_j=\sup_{k\ge j}b_k,~~\text{ and} ...
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1answer
53 views

Proof that $\liminf_{x\to\infty}f(x) \leq \limsup_{x\to\infty}f(x)$.

I can't understand this proof from my old lecture notes. $\liminf$ is defined as: \begin{align*} \liminf_{z\to\infty}f(z) = \inf_{x< y} \sup_{y< z}f(z) \end{align*} and $\limsup$ is defined ...
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1answer
51 views

Applying root test to sequence $\frac{1}{2} + \frac{1}{3} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots$

The following is an example from Principles of Mathematics, by Rudin. I've been trying to understand the example but haven't quite grasped it because it seems I can solve it differently. Given the ...
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0answers
22 views

Let $a_n$ be a bounded sequence where n is from 1 to infinite. Prove that lim sup $a_n$ and lim inf $a_n$ exist as n approaches infinite.

Let $a_n$ be a bounded sequence where n is from 1 to infinite. Prove that lim sup $a_n$ and lim inf $a_n$ exist as n approaches infinite. I'm finding difficulty in starting this proof. I have been ...