Let be the set of all subsequential limits of . Also let . So it is clear that if then . Now let us define a subsequence of as where and is strictly increasing. So since it is clear that there exists a such that .Now based on the above definition we may say that as soon as which says exactly that not only does converge, but it converges to .
That is how I would do it. Is that acceptable?
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No I don't where is the set of all subsequential limits. The same goes for
I'm not quite sure what you mean by this? Do you mean the f maps to a single value kind of constant or some other kind...either way I don't think I assumed anythingsecondly, your proof suggests that the sequence is constant, which of course, is not true for convergent sequences in general.