Why is there only one point of condensation in the set of natural numbers?

Question

higher mathematics

In the set of natural numbers, there is only one point of condensation and this is + infinity, why exactly one and how to understand this at all?
In general, as I understood, the point of thickening of the set X (for example) exists if the neighborhood of the form (a-b, a + b) contains x values ​​from X that are distinct from a. Well (b) in this definition is closely related to the limit of the sequence? b is it a small thing? , and (a) is the limit itself, and according to the idea, the number (a) should determine some sequence that converges in (a), and for example, in the set of natural numbers, can we take a sequence with a finite number? , or the main condition for the sequence of the limit is that the numbers of the variable (n) tend to infinity, just because of this fact, the set of natural numbers has only one point of condensation?

Answer 1

The point of condensation (or limit point ) of the set X (we will assume that X is a subset of the set R of real numbers) is any number P, in any neighborhood (P–d, P+d) of which there is at least one element of the set X, different from P (d is an arbitrary positive real number). Moreover, the point of condensation does not have to be an element of the set X. In the sense of this definition, the set of natural numbers N does not have any points of condensation, since for any real number S there exists a non-empty neighborhood (S–d, S+d) in which there are no one natural number different from S. If we consider N as a subset of the extended set Ř (which is obtained from R by adding to it two so-called improper numbers–/infty and +/infty, i.e. minus infinity and plus infinity), then +/infty becomes the only point of condensation for N, since in any neighborhood of this point (d, +/infty) (where d is an arbitrary real number) there are natural numbers (which, obviously, are not equal to + /infty). This follows from the fact that the set of natural numbers is not bounded from above, that is, there are arbitrarily large numbers from N. It can also be concluded that –/infty is not a condensation point for N, since, for example, the neighborhood (–/infty, 0 ) does not contain natural numbers, because natural numbers are never negative.
It should also be noted that the concept of a point of condensation of a set and the concept of the limit of a sequence in the general case do not have any direct connection. For example, you can take the sequence {0, 0, 0...}, and it will have 0 as a limit, but the set of elements of this sequence does not have condensation points (0 is no longer suitable, since there are no elements from this sets). Or you can find an example of a sequence that has no limit, but the set of elements of this sequence has two (or more) points of condensation.