How to Prove a Nonempty Set of Real Numbers is Not Sequentially Compact?

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In summary, the conversation discusses two approaches to proving that a nonempty set of real numbers, S, that is not sequentially compact has either an unbounded sequence in S or a sequence that converges to a point outside of S. The first approach involves showing that S is bounded and finding a sequence that converges to a point outside of S. The second approach involves assuming that every convergent sequence in S has a limit in S and proving that S cannot be bounded. The speaker asks if the listener has tried either approach and if they encountered any difficulties.
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atm06001
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Suppose that S is a nonempty set of real numbers that is not Sequentially compact. Prove that either (i) there is an unbounded seqeunce in S or (ii) there is a sequence in S that converges to a point x0 that is not in S.

I am having trouble with this it not being sequentially compact is screwing me up, I don't know how to prove it.
 
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Well, there are two ways you can proceed:
(1) Suppose that S is bounded, and prove that there is a sequence in S that converges to something outside of S, or
(2) suppose that every convergent sequence in S has a limit in S, and prove that S cannot be bounded.

Have you tried either way? If so, what kind of problems did you run into?
 

FAQ: How to Prove a Nonempty Set of Real Numbers is Not Sequentially Compact?

What is sequentially compactness?

Sequentially compactness is a property of a topological space that guarantees the existence of a convergent subsequence for any given sequence of points in the space.

How is sequentially compactness different from compactness?

Sequentially compactness is a weaker condition than compactness. While compactness requires that any open cover of a space has a finite subcover, sequentially compactness only requires that any sequence in the space has a convergent subsequence.

Why is sequentially compactness important in mathematics?

Sequentially compactness is important because it allows for the study and analysis of infinite objects, such as infinite sequences, in a systematic way. It also has important applications in various areas of mathematics, such as analysis, topology, and differential equations.

Can a subset of a sequentially compact space be sequentially compact?

Yes, a subset of a sequentially compact space can also be sequentially compact. This is because a subset inherits the topological properties of the original space, including sequentially compactness.

Is sequentially compactness a topological invariant?

Yes, sequentially compactness is a topological invariant. This means that if two topological spaces are homeomorphic, then they will both have the same property of sequentially compactness.

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