Let (s_n) be a bounded sequence and let S denote the set of subsequential limits

Question

Let (s_n) be a bounded sequence and let S denote the set of subsequential limits of (s_n). Prove that S is closed.

Be as thorough as possible.

Any help will be appreciated!

Explanation / Answer

Let S be the set of all sub sequential limits of a sequence {sn}. Assume that the set ~S (complement of S) is open. Now, assume x is in ~S. Let F be the image of the sequence. Now, since x is in ~S, then x is not the limit of any sub sequence. Thus, there exists a ï¥ > 0 such that a neighborhood of x with radius ï¥ has an empty intersection with F. Let y be in the neighborhood created about x. Since this neighborhood is open, the neighborhood created with respect to y must be a subset of the neighborhood of x. Again, there exists a neighborhood of y which has an empty intersection with F, which implies that y is not the limit of a sub sequence. Therefore, y must be in ~S and the neighborhoods centered about x must be a subset of ~S. We have shown that S must be closed.

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