Tagged Questions
1
vote
1answer
32 views
Suppose that for $a_n\geq b_n$ for all $n$. Show that $\varliminf_{n \to \infty} a_n\geq \varliminf_{n \to \infty} b_n$.
This is what I have so far:
Since $a_n\ge b_n$ for every $n$ then we have that $\inf\{a_n; n\ge k\} \ge \inf\{b_n; n\ge k\}$ for every $n$. When we take the limit as $n\rightarrow \infty$ we get ...
1
vote
1answer
29 views
Quick question about limits.
Sometimes, when we take limits, especially for roots and ratio tests, we define
lim of sup(a_n). Is this only because when we take the limit of sup we don't have to worry about the existence of the ...
3
votes
1answer
62 views
Can we prove $\displaystyle \limsup_{n \to \infty} \sin(n) = 1$?
Can we prove that $\displaystyle \limsup_{n\to \infty} \sin(n) = 1$?
I can prove that the above statement holds assuming that $\displaystyle \frac{\pi}{2}$ is normal (this fact is used somewhat ...
1
vote
3answers
105 views
Limit superior and inferior
How can I find the limit superior/inferior of $a_n$, as $n \rightarrow \infty $?
$$a_n=\frac{n^2+4n-5}{n^2+9}\sin^2\left(\frac{n\pi}{4}\right), n \in \mathbb N$$
I've tried Wolfram|Alpha, but it ...
4
votes
1answer
123 views
1
vote
2answers
191 views
$\limsup c_n\le \max(\limsup a_n,\limsup b_n)$
have a question that im stuck on here
Let $a_n, b_n$ and $c_n$ be three sequences of real numbers.
Suppose $k_n \in [0,1]$ for all $n$. Let $$c_n = (k_n)(a_n) + (1-k_n)b_n\;.$$ Assuming that ...
1
vote
1answer
76 views
Proof of limit inequality
Prove that for any sequence $\{x_n\}$ of positive real numbers
$$\lim\text{sup}\sqrt[n]{x_n}\leq \lim\text{sup}\frac{x_{n+1}}{x_n}.$$
My attempt:
Let $A = \lim\text{sup}\frac{x_{n+1}}{x_n}$. ...
1
vote
0answers
52 views
Probability measures on $\mathbb{T}$ whose Fourier coefficients tend to 1
Let $\mu$ be a probability measure on the complex unit circle $\mathbb{T}$. Are the following two assertions equivalent?
$\limsup_{n\to\infty}|\hat{\mu}(n)|=1$.
There exists an increasing sequence ...
2
votes
3answers
156 views
Proving that $\limsup a_n$ and $\liminf a_n$ are subsequential limits of $\{a_n\}$
Prove that if $\{a_n\}$ is a sequence, then $\limsup a_n$ and $\liminf a_n$ are subsequential limits of $\{a_n\}$.
I don't know the case where $\limsup a_n = \infty$.
0
votes
2answers
136 views
Proof correctness of: $\limsup a_n = \infty \implies \exists{a_{n_k}}$ such that $a_{n_k} \to \infty$
Prove: $\limsup a_n = \infty \implies \exists{a_{n_k}} (a_{n_k} \to \infty)$
Is this correct?
Proof:
Consider $a_n^* =\inf\{a_k : k \geq n\}$. Thus, $a_n^*$ is monotone non-decreasing. $\limsup a_n ...
1
vote
2answers
559 views
Proofs with $\limsup$ and $\liminf$
I am stuck on proofs with subsequences. I do not really have a strategy or starting point with subsequences.
NOTE: subsequential limits are limits of subsequences
Prove: $a_n$ is bounded $\implies ...
0
votes
1answer
186 views
How to prove these inequalities in real analysis?
The inequalities are:
$$\liminf(a_n + b_n) \leq \liminf(a_n) + \limsup(b_n) \leq \limsup(a_n + b_n)$$
5
votes
0answers
280 views
Proof of $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$
Let $a_n>0$ and $b_n\geq 0$, then $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$
My attempt at a proof is as follows. Let $A_n=\sup\{a_n, a_{n+1},...\}$, $B_n=\sup\{b_n, b_{n+1},...\}$, and ...
1
vote
1answer
211 views
Proof that if $s_n \leq t_n$ for $n \geq N$, then $\liminf_{n \rightarrow \infty} s_n \leq \liminf_{n \rightarrow \infty} t_n$
This is half of Theorem 3.19 from Baby Rudin. Rudin claims the proof is trivial. What I've come up with so far doesn't seem trivial, however, and is probably also wrong (my problem with it is pointed ...
3
votes
3answers
317 views
Limit superior and inferior reference request.
Is there any book where I can find theory on the limit superior and limit inferior, plus a nice deal of excercises in the same spirit as Spivak's type of excercises?
1
vote
1answer
224 views
Bounded $\limsup$ integral implies $\limsup$ bounded almost everywhere?
Consider $z \in \mathbb{R}^n$ and $\{ z_i \}_{i=1}^{\infty}$ with $z_i \rightarrow z$.
Let $\phi: \mathbb{R}^n \times X \rightarrow \mathbb{R}_{\geq 0}$. $X$ is unbounded.
I'm wondering if
$$ ...
0
votes
0answers
95 views
$\limsup$ bounded almost everywhere
Consider $z \in \mathbb{R}^n$ and a sequence $\{ z_i \}_{i=1}^{\infty}$ such that $z_i \rightarrow z$.
Let $\phi: \mathbb{R}^n \times X \rightarrow \mathbb{R}_{\geq 0}$. $X$ is unbounded.
I wonder ...
0
votes
1answer
513 views
Intuition behind $\limsup$ and $\liminf$ for probabilities
I've come across these limits in Fatou's lemma, this got me massively confused.
I'd be grateful if someone could explain the intuition behind limit suprema and limit infima of probabilities (or ...
4
votes
1answer
293 views
limit superior of a sequence proof
Let $(x_{n})\in\mathbb{R}^{+}$ be bounded and let $x_{0}=\lim\sup_{n\rightarrow\infty}x_{n}$. $\forall\epsilon>0$, prove that there are infinitely many elements less than $x_{0}+\epsilon$ and ...
1
vote
3answers
608 views
Interpretation of {Infinitely Often} = {Almost Always}
I am trying to better understand what it means for a sequence $A_n$ of subsets of a set $S$
to be such that
$\bigcap^\infty_{n = 1} \bigcup^\infty_{m = n} A_n = \limsup A_n = \liminf A_n = ...
0
votes
2answers
124 views
limit superior question
Suppose that you can apply the Ratio Test to $\Sigma a_{n}$. Let $r$ be the limit of $|a_{n+1}|/|a_{n}|$. Show that $\lim\sup|a_{n}|^{1/n}=r$ as $n\rightarrow\infty$.
I know by definition of lim sup ...
1
vote
1answer
282 views
Limit of sequence of sets - Some paradoxical facts
I am particularly confused with alternative formulas describing the inner and outer limits of a sequence of sets in topological spaces. The inner limit of a sequence of sets ...
2
votes
2answers
467 views
liminf and limsup properties
First we introduce the following notation:
$$
\mathcal{N}_\infty:= \{N\subset \mathbb{N}| \mathbb{N} \text{\ }N \text{ is finite}\}
$$
and
$$
\mathcal{N}_\infty^\#:= \{N\subset \mathbb{N}| N ...
3
votes
2answers
166 views
liminf in terms of the point-to-set distance
Let $\mathcal{X}$ be a normed space and $C\subseteq \mathcal{X}$. We define the point-to-set distance for the set $C$ to be:
$$
d_C:\mathcal{X}\ni x \mapsto d_c(x):= \inf_{y\in C}\|x-y\| \in ...
2
votes
3answers
286 views
limit superior of a sequence - showing an alternate definition
I am wondering if anybody can help me with a problem regarding the definition of the limit superior of a sequence - or rather showing an alternate but equivalent defintion holds.
The question is: The ...
9
votes
4answers
1k views
Intuitive meaning of Limit Supremum?
I am trying to understand the difference between the following two equations:
$$\bar{P} = \limsup_{t \to \infty}\frac{1}{t} \sum_{\tau = 0}^{t-1}E\{P[\tau]\} < \infty$$
and
$$\bar{P} = \lim_{t ...
1
vote
4answers
411 views
How to Quantify $\limsup \limits_{i \to \infty} \; E_i = \bigcap \limits_{k=1}^{\infty} \bigcup \limits_{i=k}^{\infty}\; E_i$?
$$\limsup_{i \to \infty} \; E_i = \bigcap_{k=1}^{\infty} \; \bigcup_{i=k}^{\infty} \; E_i$$
So
$$ \bigcup_{i=k}^{\infty} \; E_{i} = \{x \in E_{i} \mid i \in I\}= S $$
where $I$ is some index set. When ...
1
vote
2answers
536 views
Proof: Limit superior intersection
How to prove $\limsup(\{A_n \cup B_n\}) = \limsup(\{A_n\}) \cup \limsup(\{B_n\})$? Thanks!
0
votes
1answer
822 views
limit superior and limit inferior of the given sequence of sets
A sequence of sets is defined as $A_n=\{x \in [0,1] : |\sum_{i=0}^{n-1} 1_{[\frac{i}{2n},\frac{2i+1}{4n})} - 1_{[\frac{2i+1}{4n},\frac{i+1}{2n})}| \geq p\}$ for some positive $p\geq0$. What is ...
5
votes
3answers
2k views
limit inferior and superior for sets vs real numbers
I am looking for an intuitive explanation of $\liminf$ and $\limsup$ for sequence of sets and how it corresponds to $\liminf$ and $\limsup$ for sets of real numbers. I researched online but cannot ...
