I went through the proof of the question on this page: Prove $\limsup\limits_{n \to \infty} (a_n+b_n) \le \limsup\limits_{n \to \infty} a_n + \limsup\limits_{n \to \infty} b_n$ and it makes sense. I would just like to know that if an and bn have finite many terms would the proof still hold? Also if the proof holds please provide an example of strict inequality. Please pardon what is below I am using stackexchange for the first time so it isn't so neat. Let n ∈ of the Natural numbers, be defined on the interval [1,k] where ak is the final term of the sequence an. Likewise let q lie on this interval and bq is the final term of the sequence bn. Please prove that lim supn→k(an+bn)≤lim supn→k(an)+lim supn→q(bn), If the proposition is true then please provide an example of strict inequality.
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1How do you define $\limsup$ for a finite sequence ? – TheSilverDoe Jul 29 '22 at 10:31
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In my mind it would be the same as for an infinite sequence, except the set of supremums from which one takes the infimum to obtain the limsup of the sequence would always be finite. – Student Jul 29 '22 at 10:48
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If you set $\limsup u_n = \lim_{n \rightarrow +\infty} \sup_{k \geq n} u_k$, then it has no sense since $\sup_{k \geq n} u_k$ is not defined for $n$ sufficiently large for a finite sequence. – TheSilverDoe Jul 29 '22 at 10:54
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Instead of n approaching infinity wouldn't n approach the highest cardinal number in the set of supremums? – Student Jul 29 '22 at 10:59
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Please give a precise definition of what you mean by editing your question. – TheSilverDoe Jul 29 '22 at 11:18
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I have edited it – Student Jul 29 '22 at 11:44
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I would say, if $(a_1,\dots,a_N)$ is a finite sequence, then the $\limsup$ of the sequence is the last term, $a_N$. – GEdgar Jul 29 '22 at 11:47
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Makes sense now thanks – Student Jul 29 '22 at 12:15
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The proposition would not be true however if the final term of the sequence bn was negative and ak is positive, and q does not equal to k. – Student Jul 29 '22 at 20:16
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So there is strict equality if and only if k=q. – Student Jul 29 '22 at 20:28