# Questions tagged [limsup-and-liminf]

For questions concerning the definition and properties of limit superior and limit inferior of sequences of sets or real numbers.

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### Limit Superior and Inferior of a Sequence [closed]

Let Pn be a bounded sequence of real numbers and let p∈R be such that every convergent subsequence of Pn converges to p. Prove that the sequence Pn converges to p.
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### Convergence of a sequence of sets $A_n:=\{1+ \frac{m^2}{n^2}: m \in \mathbb{N} \}$

Let $n \in \mathbb{N}$ and let $A_n$ be the set given by $$A_n:=\left\lbrace 1+ \frac{m^2}{n^2}: m \in \mathbb{N} \right\rbrace$$ Can you help me to determine if $\lim_{n \to \infty} A_n$ exists? ...
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### Prove that $A_n \cap B_n \rightarrow A \cap B$

If $A_n \rightarrow A$ and $B_n \rightarrow B$ are sequences of sets then is it true that $A_n \cap B_n \rightarrow A \cap B$? How to prove or provide a counterexample? I had thought the following ...
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### $\limsup$ and $\liminf$ of family of sequences with given properties

This is from a practice exam for my quals. Let $(x_n)_{n=0}^\infty$ be an arbitrary sequence in $\mathbb R$ satisfying: $x_n \ge 0$ $x_n + 2x_{n+1} \le 6$ for all $n \ge 0$ Part $b)$ is ...
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### A seemingly true claim?

Let $(a_n)$ be a sequence of positive real numbers. If $(a_n)$ is not bounded then $\lim\sup\Big(\frac{a_n}{1+a_n}\Big)=1$. I was tempted to make the above claim when I tried proving the result below:...
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### If $f(s) = g(\frac{1}{s})$, then $\limsup\limits_{s \rightarrow \infty} f(s) = \limsup\limits_{m \rightarrow 0+} g(m)$. [closed]

Assume that $f$ is a function from $(0, \infty)$ to $\mathbb{R}$. Then we can define a function $g : (0, \infty) \rightarrow \mathbb{R}$ satisfying $f(s) = g(\frac{1}{s})$. In this case, the ...
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### Equivalence of a lim sup of functions with a lim inf of sets.

$\{\omega\in\Omega : \limsup_{n\to\infty}\lvert Y_n(\omega)-Y(\omega)\rvert=0\}$ =$\liminf_{n\to\infty}\{{\omega\in\Omega:\lvert Y_n(\omega)-Y(\omega)\rvert\lt \epsilon\}}\forall\epsilon>0$. How ...
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### Probability of Brownian Motion

I've studied a proof from the book An Introduction to Stochastic Differential Equations concerning the nowhere differentiability of the Brownian motion and I'm stuck at the following proof: In the ...
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### Proof that $a_n \leq b_n$ implies that $\limsup a_n \leq \limsup b_n$.

I am trying to prove that for sequences $(a_n)$ and $(b_n)$, that if $a_n \leq b_n$ for all $n \geq m$, then $\limsup\limits_{n \to \infty} a_n \leq \limsup\limits_{n \to \infty} b_n$. This is part ...
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