# Tagged Questions

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

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### $\lim \sup\{X_n\geq x\}$ vs $\{\lim \sup X_n \geq x\}$

Let $(X_n)$ (n is a natural number) be a sequence of real valued random variables. For any real number $x$, let's define: $E_x = \limsup \{ X_n \geq x\}$, $F_x = \{\limsup X_n \geq x\}$ If $x$ is ...
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### Proving a lower bound on the limit superior of a sequence.

Prove that for every positive sequence {$a_{n}$}, $$\varlimsup_{n \to \infty}\frac{\sum_{i=1}^{n+1}a_{i}}{a_{n}}\geq 4$$ Also find the sequences {$a_{n}$} for which 4 is attained. Attempted ...
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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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### Relations and differences between outer/inner limit and Kuratowski limsup/liminf

Let $X$ be a topological space. I am asking about the relations and differences between the following two different types of $\limsup$ and $\liminf$ of $A_n ⊆ X, n ∈ \mathbb{N}$, a sequence of ...
### $\limsup$ 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 ...