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

Let {s_{i}} and {t_{i}} be bounded sequences of real numbers, let E, F, G be the sets of all subsequential limit points of {s_{i}}, {t_{i}}, {s_{i}t_{i}} respectively. Prove that [tex]G\subseteq EF= \left\ ef|e\in E,f\in F \right\[/tex]

2. Relevant equations

3. The attempt at a solution

I have trouble understanding the behavior of the sequence s_{i}t_{i}. For this question, it seems that the right thing to do is to choose any x in G, show that x is in EF. Here's what I've done:

There is a subsequence [tex]s_{\alpha _{i}}t_{\alpha_{i}}[/tex] that converges to x. Now, consider the sequences [tex]{s_{\alpha _{i}}}[/tex] and [tex]{t_{\alpha _{i}}}[/tex]. Being a sequence in the compact set (i.e. [-M,M] in R), some subsequence of [tex]{s_{\alpha _{i}}}[/tex] converges to a point a, let A be the set of these subscripts. And some subsequence of [tex]{t_{\alpha _{i}}}[/tex] converges to a point b, Let B be the set of these subscripts.

If [tex]A\cap B[/tex] has infinitely many elements, then [tex]x=ab\in EF[/tex]. But what if [tex]A\cap B[/tex] is empty? I've no idea why this cannot happen. Any hint would be greatly appreciated! Is there any different way to prove it?

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# Homework Help: Subsequencial limit set of a product sequence SiTi

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