(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let (sn) be a sequence inRthat is bounded but diverges. Show that (sn) has (at least) two convergent subsequences, the limits of which are different.

2. Relevant equations

3. The attempt at a solution

I know that a convergent subsequence exists by Bolzano-Weierstrass. I'm having trouble showing that multiple convergent subsequences exist, and how to show that they would have different limits. I know that (sn) will probably look something like (-1)^n (so it will be a sequence that neither increases or decreases, but rather most likely "bounces around"), and if I had an actual sequence to prove this on I could probably do it, but to prove it in general is giving me some difficulty.

The only thought I had was that maybe defining two subsequences of (sn) that have only one element, i.e. (an) = {s1} and (bn) = {s2}. I'm not sure this would work, however, since I am not sure that an infinite constant sequence, such as (an), would be a subsequence of (sn), seeing as how, (s1) would occur much fewer times in (sn) than it would in (an) (for example: (sn) = sin(n), (an) = sin(1), then (an) = {sin1, sin1, sin1, sin1, ...} and (sn) = {sin1, sin2, sin3, sin4, ...}).

Any help would be greatly appreciated.

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# Bounded sequence that diverges, convergent subsequence

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