One version of the Bolzano-Weierstrass theorem states that:"An infinite bounded set of real numbers has at least one limit point." From the definition of limit point, it's easy to see that a finite set can not have a limit point. However, the requirement of boundedness is not so obvious. Before giving the proof of the theorem, my textbook uses the set of integers as an example, says that "the set of integers, though infinite, has no limit point because it's not bounded." How to understand this sentence?
[Math] Why are both of the conditions of Bolzano -Weierstrass theorem necessary
real-analysis
Best Answer
Clearly, in order to have any kind of limit point, you need an infinite number of elements. But just because you have an infinite number of elements, doesn't mean you'll have a limit point, which is what the example you mentioned shows: $\mathbb Z$ is infinite and yet has no limit point.
The Bolzano-Weierstrass theorem basically says that bounded sets don't have enough "room" to be able to hold an infinite number of points without those points clustering around some limit point.