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.

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 […]

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

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 […]

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 […]

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 […]

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 […]

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 […]

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

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: $$ \mathcal{P}^1(S) := \{P: \mathcal{B}_S \rightarrow [0,1] \mid P \mbox{ probability measure:} \forall a\in S: \int d(a,x) P(dx) < \infty \}, $$ […]

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