My question has to do with the completeness property of the group R (real numbers)
It seems to be a basic question but I have no idea how to formally prove what i need.
I need to prove that infimums and supremums exist.
Let there be group A(a,b) which is a subgroup of R. Prove that group A has a supremum and an infimum. Meaning - Prove that there is a smallest upper bound, and largest lower bound.
I know they exist, but don't know how to prove it
I just started university like 2 weeks ago, so I'm not yet familiar with all the terms and everything. The lecturer asked this during infinitesimal calculus class, so i figured it is related to it.
How do I know infimum and supremum exist? logic. There has to be a barrier, a value that if u go past him (be it to the left or to the right) even by the slightest epsilon, you entered the group. there has to be like a limit.
By R I mean all the rational numbers and irrational numbers. I mean this Real number - Wikipedia, the free encyclopedia
This depends very much on exactly how the real numbers are defined. Just saying "the rational and irrational numbers" is not enough. For example, many texts use the "Dedekind Cut" definition of the real numbers specifically because it makes proving that every bounded set of real numbers has an infimum and extremum very easy.
Also, far as I know, that has little to do with "infinitesmals". Also, you say "Let there be group A(a,b) which is a subgroup of R. Prove that group A has a supremum and an infimum. Meaning - Prove that there is a smallest upper bound, and largest lower bound." There may be a translation problem here. The set of a all integer is a "subgroup" of R but does NOT have a supremum and infimum. Perhaps you mean simply "subset" but even then you have to add "bounded" and I don't see that here. I don't see how we can help you if we don't know what definition of "real numbers" you are using and what you actually know about "infimum" and "supremum".