Question: a) A subset E of R is said to be sequentially compact if and only if every sequence xn E has a convergent subsequence

a) A subset E of R is said to be sequentially compact if and only if every sequence xn ∊ E has a convergent subsequence whose limit belongs to E. Prove that every closed bounded interval is sequentially compact.
b) Prove that there exist bounded intervals in R that are not sequentially compact.
c) Prove that there exist closed intervals in R that are not sequentially compact.

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a To show E a b is sequentially compact let x n E By the Bolzano Weierstrass Theorem x n ... View full answer

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