matt grime said:It's slightly nicer to show that a bounded set in R cannot be dense. Many proofs by contradiction are unnecessary - i.e. you wish to show A implies B, so you assume A and not B and show not B implies not A, without any use of the assumption of A.
matt grime said:What? The point was just make a constructive proof of the contrapositive statement without making an unnecessary preliminary assumption.
A set A is considered dense in Rn if every point in Rn can be approximated by a point in A. A set is considered unbounded if it has no finite bound or limit. Therefore, "A Dense in Rn Not Bounded" means a set in Rn that contains points that can approximate any point in Rn, but has no finite bound or limit.
Studying "A Dense in Rn Not Bounded" is important because it helps us understand the properties and behavior of unbounded sets in mathematical spaces. It also has applications in various fields such as topology, analysis, and geometry.
The main difference between "A Dense in Rn Not Bounded" and "A Dense in Rn" is that the former does not have a finite bound or limit, while the latter may have a finite bound or limit. In other words, "A Dense in Rn Not Bounded" contains points that are infinitely far apart, while "A Dense in Rn" may contain points that are only a finite distance apart.
Yes, it is possible for a set to be both dense and bounded in Rn. For example, the set of rational numbers between 0 and 1 is both dense (every real number between 0 and 1 can be approximated by a rational number) and bounded (the set has a finite limit, 1).
Some real-world examples of "A Dense in Rn Not Bounded" include the set of all real numbers (which is unbounded) and the set of all irrational numbers (which is dense in the set of real numbers). These sets have applications in various fields such as physics, engineering, and economics.