Articles of limsup and liminf

Proof of $\limsup\sin nx=1, n\rightarrow \infty \forall x\in \mathbb{R}.$

I have a problem: need to prove $$\limsup_{n\to\infty}\sin nx=1$$ for all x without set which measure is equal zero. We need to come up with a sequence with limit that equals to one, but I don’t know how to do this.

Theorem 3.17 in Baby Rudin: Infinite Limits and Upper and Lower Limits of Real Sequences

Here’re Definitions 3.15 and 3.16 and Theorem 3.17 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. Definition 3.15: Let $\{s_n \}$ be a sequence of real numbers with the following property: For every real $M$ there is an integer $N$ such that $n \geq N$ implies $s_n \geq M$. We then […]

Show root test is stronger than ratio test

This question already has an answer here: Inequality involving $\limsup$ and $\liminf$: $ \liminf(a_{n+1}/a_n) \le \liminf((a_n)^{(1/n)}) \le \limsup((a_n)^{(1/n)}) \le \limsup(a_{n+1}/a_n)$ 1 answer

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 subsets of $X$. From Wikipedia $\limsup A_n$, which is also called the outer limit, consists of those elements which are limits […]

Theorem 3.37 in Baby Rudin: $\lim\inf\frac{c_{n+1}}{c_n}\leq\lim\inf\sqrt{c_n}\leq\lim\sup\sqrt{c_n}\leq \lim\sup\frac{c_{n+1}}{c_n}$

Here’s Theorem 3.37 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: For any sequence $\{c_n\}$ of positive numbers, $$\lim_{n\to\infty} \inf \frac{c_{n+1}}{c_n} \leq \lim_{n\to\infty} \inf \sqrt[n]{c_n},$$ $$ \lim_{n\to\infty} \sup \sqrt[n]{c_n} \leq \lim_{n\to\infty} \sup \frac{c_{n+1}}{c_n}.$$ Now Rudin has given a proof of the second inequality. Here’s my proof of the first. Let $$\alpha […]

Set convergence and lim inf and lim sup

I’m a bit confused with the general concept of convergence of a sequence of sets. I’m well aware that the limit of a sequence $\{C^{\nu}\}$ exists iff $$\liminf_{\nu \rightarrow \infty} C^{\nu} = \limsup_{\nu \rightarrow \infty} C^{\nu}$$ where lim inf (resp. lim sup) is the set of points that appear in the limit all but finitely […]

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: $$ d_C:\mathcal{X}\ni x \mapsto d_c(x):= \inf_{y\in C}\|x-y\| \in [0,\infty] $$ Additionally, we define the inner limit of a sequence of sets $C_n$ in $\mathcal{X}$ to be: $$ \liminf_n C_n = \bigcup_{n=1}^\infty \bigcap_{m=n}^\infty C_m $$ This […]

Showing that lim sup of sum of iid binary variables $X_i$ with $P = P = 1/2$ is a.s. infinite

Let $(X_i)_{i\in\mathbb{N}}$ be an i.i.d. sequence of binary random variables with $$P[X_i = 1]=P[X_i = -1] = \frac{1}{2}$$ and let $$S_n = \sum_{i=1}^{n} X_i.$$ I’d like to show that $$P[\lim \sup_{n \rightarrow \infty} S_n = \infty] = 1$$ with the means of basic probability theory and the Borel–Cantelli lemma or Kolmogorov’s 0-1 law. Could somebody […]

Is it true that $\limsup \phi\le\limsup a_n$, where $\phi=\frac{a_1+…+a_n}n$?

This question already has an answer here: If $\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$ then $\operatorname{{lim sup}}\sigma_n \leq \operatorname{lim sup} s_n$ 2 answers

(dis)prove:$\sup_{F \in 2^{(L^1(S,\mathbb{R}))}}\limsup\sup_{f\in F}|\int f dP_n-\int fdP|=\limsup\sup_{f\in L^1(S,\mathbb{R})}|\int fdP_n-\int fdP|$

Let $(S,d)$ be a complete separable metric space and consider the set $L^1(S,\mathbb{R})$ of functions $f:S \rightarrow \mathbb{R}$ which are 1-Lipschitz, i.e. $\forall x,y \in S: |f(x) – f(y)| \leq d(x,y)$. Further let: $$ \mathcal{P}^1(S) := \{P: \mathcal{B}_S \rightarrow [0,1] \mid P \mbox{ probability measure:} \forall a\in S: \int d(a,x) P(dx) < \infty \}, $$ […]