# Compact implies Sequentially Compact

• boopbeep
In summary, the conversation discusses the proof that if X is a metric space and E is a subset of X that is compact, then E is also sequentially compact. The conversation mentions considering a sequence x_n in E and finding a point a in E and a radius r > 0 such that Br(a) contains x_k for infinitely many k's. The conversation also clarifies that if any subsequence of x_n converges to a, then x_n also converges to a, and that this is what sequential compactness is about. However, this statement is not always true, as demonstrated by the example of a_n=(-1)^n=1,-1,1,-1,1,-1,... where a_{2n}=
boopbeep
Homework Statement [/b]
I need help proving that if X is a metric space and E a subset of X is compact, then E is sequentially compact.

I know I need to consider a sequence x_n in E, and I want to say that there is a point a in E and a radius r > 0 so that Br(a) [the ball of radius r with center a] contains x_k for infinitely many k's. If I show this, then I think I can conclude that any subsequence of x_n converges to a. Can someone please help?

So... your conjecture is that every sequence in a compact metric space is actally convergent?

No, that is not one of the assumptions...my second paragraph in the problem description is a hint in the back of the textbook as to how to go about the problem.

If any subsequence of xn converges to a, then xn converges to a too. Rather, what you can conclude is a single subsequence of xn converges to a (which is what sequential compactness is about)

Office_Shredder said:
If any subsequence of xn converges to a, then xn converges to a too. Rather, what you can conclude is a single subsequence of xn converges to a (which is what sequential compactness is about)

This is not true- consider a_n=(-1)^n=1,-1,1,-1,1,-1,...

Then a_{2n}=1 is a subsequence that converges to 1, but a_n does not converge at all- nonetheless to 1.

Office_Shredder said:
If any subsequence of xn converges to a, then xn converges to a too. Rather, what you can conclude is a single subsequence of xn converges to a (which is what sequential compactness is about)

cfgauss36 said:
This is not true- consider a_n=(-1)^n=1,-1,1,-1,1,-1,...

Then a_{2n}=1 is a subsequence that converges to 1, but a_n does not converge at all- nonetheless to 1.

Office Shredder meant "if every subsequence". He chose the wrong word.

## 1. What is the definition of compact and sequentially compact?

Compact and sequentially compact are two different types of properties that a topological space can have. A space is considered compact if every open cover has a finite subcover, meaning that a finite number of open sets can cover the entire space. On the other hand, a space is said to be sequentially compact if every sequence in the space has a convergent subsequence.

## 2. How are compact and sequentially compact related?

Compact and sequentially compact are closely related properties, with compactness being a stronger condition than sequential compactness. This means that every sequentially compact space is also compact, but the converse is not necessarily true.

## 3. What is the Compact implies Sequentially Compact theorem?

The Compact implies Sequentially Compact theorem states that if a topological space is compact, then it is also sequentially compact. In other words, if a space satisfies the condition of compactness, then it automatically satisfies the condition of sequential compactness.

## 4. What are some examples of compact spaces that are not sequentially compact?

There are several examples of compact spaces that are not sequentially compact, such as the space of continuous functions on a closed interval with the supremum norm, or the space of bounded real sequences with the supremum norm. These spaces are compact but not sequentially compact because they do not satisfy the Bolzano-Weierstrass property, which is a necessary condition for sequential compactness.

## 5. How is the Compact implies Sequentially Compact theorem useful in mathematics?

The Compact implies Sequentially Compact theorem is an important result in mathematics, particularly in the field of topology. It allows us to simplify proofs and make connections between different concepts, as well as providing a useful tool for proving other theorems. Additionally, many important theorems and results in mathematics rely on the compactness property, making the Compact implies Sequentially Compact theorem a powerful tool in various areas of mathematics.

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