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Prove, by nonstandard reasoning, that the limit superior of a sequence is a cluster point.

A natural approach to try would be to use the ideas around saturation/idealisation. With respect to the order relation, finding the maximum among the extended terms of the sequence is obviously ...
Mikhail Katz's user avatar
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Prove, by nonstandard reasoning, that the limit superior of a sequence is a cluster point.

$ \newcommand\ext{{}^*} \newcommand\N{\mathbb N} \newcommand\sh{\mathop{\mathrm{sh}}} $Let $\N_\infty = \ext\N\setminus\N$ and let $S = \{\sh(\ext s_\omega) \;:\; \omega \in \N_\infty\}$ be the set of ...
Nicholas Todoroff's user avatar
1 vote

Prove, by nonstandard reasoning, that the limit superior of a sequence is a cluster point.

I couldn't quite reconcile my understanding of Goldblatt with the details provided in the other answers. Given my intense struggle over this same question and eventually figuring out a proof, I ...
J. Chapman's user avatar
1 vote

intuition on lim sup and lim inf in probability spaces

The terminology "almost all" in this situation simply means all but finitely many (also said cofinitely many). It does not really have anything to do with the order structure of the natural ...
C. icosahedra's user avatar
2 votes
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intuition on lim sup and lim inf in probability spaces

I don't know whether I can give a better explanation of the intuition than you have already given, but consider the following example. $$ A_n = \emptyset \qquad n \text{ even} \\ A_n = \Omega \qquad n ...
Chris's user avatar
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2 votes
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$\lim \sup$/ $\lim \inf$ inequalities fail when $\mu(X)=\infty$

Here's an easy example; consider the sets $$ A_n=[n,\infty) $$ Then, $$ \lim\sup A_n=\bigcap_{n=1}^\infty\bigcup_{k=n}^\infty A_k=\bigcap_{n=1}^\infty\bigcup_{k=n}^\infty [k,\infty)=\bigcap_{n=1}^\...
Josh B.'s user avatar
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1 vote
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$\inf_{n\in \mathbb{N}^*} \left\{\frac{3^n}{2^n} \right\} > 0$?

I would expect this to be false - the powers of a real number mod 1 should be uniformly distributed on [0, 1] unless there's some reason to expect otherwise. The particular case of the powers of $3/2$...
Michael Lugo's user avatar
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