# Tagged Questions

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### Integrating the two-views of lim sup and lim inf

Preliminaries Let $(x_n)$ be a bounded real number sequence and $(x_n )_{n≥k}$be a subsequence of $x_n$ which only takes the values of the sequence starting from the k−th term. Let {$x_n$} ...
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### Is liminf of a product equal to the product of liminfs?

My question is just for curiosity. I was thinking if is true this curious affirmation: Let $a_n$ a bounded sequence of nonnegative numbers and $b_n$ a convergent sequence of negative numbers. Then ...
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### Bounded limit inferior and limit superior of functions on dense sets implies divergence on uncountable dense set.

Reading an article by Akcoglu et al. "The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters", which can be found here: ...
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### Question about the definition of $\limsup$ and $\liminf$ on real valued functions

I understood the definition of $\liminf$ and $\limsup$ for sequences well. But there is a little bit of confusion when it comes to functions. If I did learn it correctly, the definition of ...
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### Problems understanding definition of limit superior.

I'm aware that there are multiple questions on this topic and I have read most of them, but I still don't really understand the definition and would really appreciate some quick help with this, ...
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### Is my proof about $\liminf$ of bounded real sequences correct?

Yesterday I asked in the question at A proof about the limit infimum of a bounded sequence about the proof of the following statement: Let $x_n$ be a bounded sequence of real numbers. Then the ...
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### A proof about the limit infimum of a bounded sequence

I tried to find a proof for this statement: If we have a bounded sequence $x_n$, then the limit infimum is defined as $a=\liminf_n x_n$ such that $a$ is the largest of real numbers which have the ...
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### The definitions of limit infimum and limit supremum

I have begun reading Rosenthal's "A First Look to Rigorous Probability Theory" and in order to reinforce my calculus background I am studying through the appendix section. Here he defined the limit ...
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### Subsequences and limit inferior

Suppose $f\colon \mathbb{R} \to \mathbb{R}$ is a continuous function. Let $x\in \mathbb{R}$ and let $(x_n) \subseteq \mathbb{R}$ be a sequence converging to $x$. Let $(y_n)$ be a subsequence of ...
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### Limsup of intervals $(-\infty, - n \sin(1/n))$

I think that $\limsup \big (-\infty, - n \sin \frac{1}{n} \big )$ (as the lim sup of sets) is $(-\infty, 1)$. Is this correct? If so, how do we prove it? Background: I'm trying to do a question in ...
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### Question of double limits

I post a problem concerning a possible generalization of the question of interchanging in double limits. Given a sequence of functions $\{f_{j}\}_{j}$ on an interval $I$ and a point $a\in I$, is it ...
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### Is this an increasing sequence?

From this beginning of this document - Let $(c_n)$ be a bounded sequence of real numbers. Deﬁne $a_n = \inf\{c_k : k ≥ n\}$ The sequence $(a_n)$ is bounded and increasing so it has a ...
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### Questions on limit superiors

Given any bounded sequence $x(n)$, let $l$ be its limit superior. Let $\epsilon > 0$. Does there exist $n \in \mathbb{N}$ such that $x(n)<l+\epsilon$? Does there exist infinitely many $n$ such ...
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### Show that if $\lim_{k \rightarrow \infty} \sup (\sqrt[k]{|a_k|}) < 1$ then $\sum_{k=0}^\infty a_k$ converges absolutely.

Let $(a_k)_{k \in \mathbb{N}}$ be some real sequence. Show that if $\lim_{k \rightarrow \infty} \sup (\sqrt[k]{|a_k|}) < 1$ then $\sum_{k=0}^\infty a_k$ converges absolutely. I have some general ...
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### Show that if $(a_k)_{k \in \mathbb{N}}, (b_k)_{k \in \mathbb{N}}$ are bounded, then $\lim \sup(a_k + b_k) \leq \lim \sup(a_k)+\lim \sup (b_k)$

Show that if $(a_k)_{k \in \mathbb{N}}, (b_k)_{k \in \mathbb{N}}$ are bounded, then $\lim \sup(a_k + b_k) \leq \lim \sup(a_k)+\lim \sup (b_k)$ I'm not quite sure how to prove this. I want to try ...
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### What does $\liminf_{n\to \infty} f_n$ and $\limsup_{n\to \infty} f_n$ mean?

I have searched the internet, including this website, and I have found some answers, but none which I understand what actually mean. What is $\liminf_{n\to \infty} f_n$? Is it a single value? Is it a ...
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### $\inf$ of a sequence of $\sup$

let $f_n$ a sequence of right-continuous functions $[0,\infty) \to R$. Assume that for each $T$: \begin{align} & \sup_{\substack{t \in [0,T]\\n \in N}} |f_n(t)|<\infty \end{align} Is it true ...
### $\lim\sup(|s_n|) = 0$ iff $\lim(s_n) = 0$
I need to prove that $\lim\sup(|s_n|) = 0$ if and only if $\lim(s_n) = 0$. I totally get the intuition behind this implication but not sure how to give a formal proof of it. Here is my intuition: for ...