What Are the Key Differences Between Complete and Sequentially Compact Spaces?

[Hymne]

Hello Physicsforums!
I have a problem with the difference between complete metric space and a sequentially compact metric space.
For the first one every Cauchy sequence converges inside the space, which is no problem.
But for the last one "every sequence has a convergent subsequence." (-Wiki) And it's here that I get lost.

How does this affect the constraints on the space?
Could someone please try to give me an intuitive explanation?

For [1,9] on the real axis we can take the sequence (1,2,3,4,5,6) as an example. How do we find a convergent subsequence in this one?
Have I missunderstood it all?

 

[George Jones]

For [1,9] on the real axis we can take the sequence (1,2,3,4,5,6) as an example. How do we find a convergent subsequence in this one?
Have I missunderstood it all?

What is the definition of "sequence"?

 

[Hymne]

What is the definition of "sequence"?

Hmm, I use this one http://en.wikipedia.org/wiki/Sequence .
With

In mathematics, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and the exact same elements can appear multiple times at different positions in the sequence.

Maybe it´s here that I am confused.

Should we only work with Cauchy sequences maybe?

 

[Werg22]

These definitions apply to infinite sequences. (1,2,3,4,5,6) is not an infinite sequence. It doesn't even mean anything for a finite sequence to converge!

 

[jav]

To the original question..

In a complete metric space (a_n) converges <-> (a_n) is cauchy

In a compact metric space, every sequence a_n contains a convergent subsequence (a_nk).

We should note that convergence -> cauchy in any metric space.

Then, in a compact metric space, every sequence a_n contains a cauchy subsequence (a_nk).

Regardless, the properties of these two types of spaces are completely different.

A simple example highlighting the difference between the two is a subset of R¹. Consider, the interval (0,1).

By the Heine-Borel theorem, this space is not compact since it is not closed.

It is, however, a complete metric space since cauchy <-> convergent in R¹.

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