Analysis - compactness and sequentially compactness

[quasar987]

whoopsies

 

[HallsofIvy]

1. Homework Statement
According to the dfn, a subset A of a metric space is sequentially compact is every sequence in A has a subsequence that converges to a point in A.

An example of a sequentially compact set that comes to mind is R itself.

Then, Bolzano-Weierstrass's thm says that sequentially compactness and compactness are equivalent.

Finally, Heine-Borel's thm says that in R^n, compactness and closed+bounded are equivalent.

Thus, in R^n, closed+bounded and sequentially compactness are equivalent. But R is not bounded. What's going on?

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?