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## Homework Statement

Serge Lang

*Undergraduate Analysi*s Chapter Ⅷ §1 Exe4

Let{Xn} be a sequence in a normed vector space E such that {Xn} converges to v. Let S be the set consisting of all v and Xn.

Show that S is compact.

## Homework Equations

None

## The Attempt at a Solution

I guess that maybe it is useful to consider it from the aspect of the definition of compactness,i.e. every sequence of elements of S has a convergent subsequence whose limit is in S. But I coudn't convince that why there must be such a convergent subsequence in every sequence, you know, some sequences are not the given ones that converge.

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