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whoopsies 
Why would R "come to mind" as an example of a sequentially compact set? In particular what do you say the sequence {n}= {1, 2, 3, 4,...} converges to?quasar987 said:
Compactness and sequentially compactness are two important concepts in mathematical analysis that measure how "small" a set is. A set is considered compact if it is closed and bounded, meaning it contains all its limit points and can be contained in a finite-sized ball. Sequential compactness refers to the property that every sequence in the set has a convergent subsequence.
Both compactness and sequential compactness are measures of "smallness" in a set, but they differ in their definitions. Compactness is a topological property that depends on the set's open and closed sets, while sequential compactness is a metric property that depends on the set's distance function. Additionally, compactness is a stronger condition than sequential compactness, as all compact sets are also sequentially compact, but not vice versa.
Compactness is important in mathematical analysis because it allows for the study of functions and sequences in a more structured and well-behaved space. It guarantees the existence of important mathematical objects, such as global extrema and solutions to differential equations. Compactness also allows for the use of powerful theorems, such as the Heine-Borel theorem and the Bolzano-Weierstrass theorem.
Compactness can be used to prove the convergence of a sequence by showing that the sequence is contained in a compact set. Since compact sets are closed, the sequence must also be contained in the set's closure. By the Bolzano-Weierstrass theorem, the sequence must have a convergent subsequence, and since the subsequence is contained in the original sequence, the original sequence must also converge.
Yes, a set can be compact and not sequentially compact. This is because compactness and sequential compactness are two different properties that measure different aspects of a set. A set can be compact if it is closed and bounded, but it may not have the property that every sequence in the set has a convergent subsequence. An example of this is the set of real numbers with the usual metric, which is compact but not sequentially compact.