Timeline for Having trouble proving quasi-increasing sequence proof.
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 20, 2021 at 6:40 | vote | accept | Mobius.Drip | ||
| Jul 20, 2021 at 1:18 | comment | added | Mobius.Drip | @AlejandroRuiz see answer below. | |
| Jul 20, 2021 at 1:17 | answer | added | Mobius.Drip | timeline score: 5 | |
| Jul 19, 2021 at 19:54 | comment | added | Alex | I would like to know if you got to solve this problem. Kind of struggling myself with it right now. | |
| Feb 14, 2021 at 1:32 | comment | added | Aphelli | Okay... then you can prove “directly” that the sequence is Cauchy. | |
| Feb 14, 2021 at 1:28 | comment | added | Mobius.Drip | I see I looked in the book and the limit point is defined after this chapter in the topology section so I have yet to learn such a definition. So I suppose I need to use Bolzano-Weierstrass, claim there are two different limits and arrive at a contradiction? | |
| Feb 14, 2021 at 1:25 | comment | added | Aphelli | Do you know what a limit point is? (Not a limit – so it’s certainly not enough to show that any two limits should be the same. You need to prove that the limits of any two convergent subsequences must be the same). If so, the limsup is the supremum of the limit points. In other words, it’s the smallest real number $r$ such that for any $\epsilon>0$, the sequence contains finitely many terms above $r+\epsilon$. Or the limit of the decreasing sequence $\left(\sup_{p\geq n}\,{a_p}\right)_n$. | |
| Feb 14, 2021 at 1:00 | comment | added | Mobius.Drip | I see, thank you. I am wondering If I am allowed to claim it converges since that is what we are trying to show. Or should I argue by contradiction that if two limits exist they would be the same? Im having trouble understanding what $\lim\sup$ means since $\sup(a_n)$ is just a single value. | |
| Feb 14, 2021 at 0:53 | comment | added | Aphelli | Actually, $a_n$ doesn’t converge to its sup, because (for instance) you can add finitely many very large values at the beginning and the sequence remains quasi-increasing. What $a_n$ converges to is its limsup. The simplest way to prove it, I think, is to show that a bounded quasi-increasing sequence has at most one limit point – I think you can somewhat imitate the proof of uniqueness of limits. | |
| Feb 14, 2021 at 0:46 | history | asked | Mobius.Drip | CC BY-SA 4.0 |