# Tagged Questions

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

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### The limit supremum of a function involving Brownian motion

I would like, for some $\delta>0$ and a Brownian motion $B$, to calculate $\displaystyle\limsup_{t\to\infty}\left(\exp\left( (1+\delta)t\right)\cdot\exp\left(-B_t-\frac{t}{2}\right)\right)$ ...
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### Can I conclude the following about limsup

I am trying to show that if $F:[a,b]\rightarrow\mathbb{R}$ is continuous and of bounded variation then $g(x)=\limsup_{h\rightarrow 0, h>0} \frac{F(x+h)-F(x)}{h}$ is a Lebesgue measurable function. ...
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### does taking limsup preserve inclusion relation?

Let $\{a_n\}_{n=1}^{\infty}$ and $\{b_n\}_{n=1}^{\infty}$ be sequences of sets with no further structure at that point, such that $a_n \subset b_n$ for every $n\in \mathbb{N}$, does it holds that ...
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### find Lim sup $X_n$ and Lim inf $X_n$?

Question: let $X_{n}=\frac{(n-1)(-1)^{n}}{n}$ find the $\limsup(X_{n})$ and $\liminf(X_{n})$ Can I get someone to help me with this proof? I get that $\frac{(-1)^{n}}{n}$ gives me ...
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### A doubt on limit supremum

In a book, I see the following : $\limsup_{n \to\infty} |a_n| \geq 1$ implies $\limsup_{n \to \infty} |a_n|^{\frac{1}{n}} \geq 1$. Why ?
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### sequence of sets with $\limsup A_n = \mathbb N$

Find a sequence of one-point-sets $A_n = \{\ell_n\}$ with $\ell_n\in\mathbb N$ for all $n\in\mathbb N$, such that $$\limsup_{n\to\infty} A_n=\mathbb N$$ I know the definition of the $\limsup$ of a ...
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### a problem about liminf/ limsup with a continuous function

My Mathematical Analysis III professor gave me this problem: Let $f:(0,1) \rightarrow f((0,1))$ be a continuous function in the standard euclidean metric space $($$\Bbb R,d_2$$)$ and let ...
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### Prove that $\sup_k f_k, \inf_k f_k, \lim \sup_k f_k, \lim \inf_k f_k, \lim_k f_k$ (if it exists) are all M-measurable.

If $f_1, f_2, f_3,...$ are $M$-measurable, prove that $\sup_k f_k, \inf_k f_k, \lim \sup_k f_k, \lim \inf_k f_k, \lim_k f_k$ (if it exists) are all M-measurable. My thoughts: We know for any sequence ...
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### Prove or give counterexample for the statements on limsup

Let $(a_n)$ and $(b_n)$ be two real-valued and bounded sequences. $\limsup \max \{a_n,b_n\} = \max\{\limsup a_n, \limsup b_n\}$. $\limsup\min\{a_n,b_n\} = \min\{\limsup a_n, \limsup b_n\}$. EDIT: ...
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### Limit superior inequalities proof

Let $a_n$ be a positive sequence. Prove that $$\limsup_{n\to \infty} \left(\frac{a_1+a_{n+1}}{a_n}\right)^n\geqslant e.$$
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### limit superior and limit inferior proof

$$\limsup \left(\frac 1{a_n} \right)=\frac 1{\liminf(a_n )}$$ I know this is true base on the definition of $\limsup$ and $\liminf$, but I don't know how to prove it formally.
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### Lim sup of sequence of sets and theirs unions [closed]

I have to prove the following equality: Can somebody help me to prove this?
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### Lim Sup and Lim Inf

This is a question with a few components to it. In each one, I give my attempt at a solution. Thank you in advance for any suggestions/answers/advice on how to solve this problem. For bounded ...
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### Understanding the supremum limit of a set

Given a sequence ${A_n}$, we define the set lim sup $A_n = \{x : x$ belongs to infinitely many $A_n$'s$\}$ That is - lim sup $A_n = \bigcap_{m=1}^\infty (\bigcup_{n=m}^\infty A_n)$ I can't see how ...
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### $\sup_{n >k}a_n \sup_{n>k}b_n = \sup_{m,n > k} a_nb_m \geq \sup_{n > k} a_nb_n$

I'm considering the proof written by robjohn in the following post: lim sup inequality $\limsup ( a_n b_n ) \leq \limsup a_n \limsup b_n$ Now, in that proof, he says that \sup_{n >k}a_n ...
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### Liminf and Limsup in measure theory and in sequences

In measure theory, given sets $A_1,A_2,\ldots$, we define $\liminf A_n=\bigcup_{k=1}^\infty\left(\bigcap_{n\geq k}A_n\right)$ and $\limsup A_n=\bigcap_{k=1}^\infty\left(\bigcup_{n\geq k}A_n\right)$. ...
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### Fatou's lemma for a generalized Laplacian of subharmonic functions

Given a nonnegative smooth function $g$ with compact support in the complex plane (e.g. a mollifier) and a subharmonic function $u$ (defined on a sufficiently large domain in $\mathbb C$), I would ...
My question is: Let $(x_n)$ be a bounded sequence of real numbers. Prove that for every $\epsilon > 0$ and every $N\in\mathbb{N}$ there are $n_1, n_2\geq N$ such that ...
For a sequence $X_n$ of real numbers with $\limsup\limits_n X_n = \inf\limits_n\sup\limits_{k\geqslant n} X_k$, $\liminf\limits_n X_n = \sup\limits_n \inf\limits_{k\geqslant n} X_k$, (a) How to prove ...