Proof: Minimum of a Set is the Infimum | Real Analysis

The minimum of a set is also the infimum of the set, we will prove this in today's lesson! This also applies to functions, since the range of a function is just a set of values. So if a function takes on a minimum value m, then the minimum m is also the infimum of the function.
Recall that the minimum of a set A is an element m in A such that m is less than or equal to every element a of A. The infimum of a set A is the greatest lower bound: as in, of all values that are less than or equal to every element of A, the greatest such value is the supremum, written as inf A. Not every set has a minimum or an infimum, but if a set has a minimum then that min is also the inf! We will prove this using contradiction.
Definition of supremum and infimum of a set: • Definition of Supremum...
Maximum of a set is the Supremum: • Proof: Maximum of a Se...
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[@WrathofMath]

Check out my Real Analysis playlist for more! kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@a.robberts4225]

Absolutely Amazing Presentation! Makes any viewer see Set Theory/Real Analysis in a different spotlight.

[@WrathofMath]

Thanks so much! If you're looking for more real analysis, check out my playlist: kzbin.info/aero/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

[@catherineferris4379]

Super helpful! More real analysis videos please!

[@WrathofMath]

Thanks for watching, Catherine! More real analysis videos are on the way. I am mostly going to follow the order of a typical course in the topics I cover, so naturally it will take a while to complete my playlist of Real Analysis lessons. Let me know if you ever have any requests, I'm happy to jump ahead in the course material for a viewer request.

[@kaoutherlhb306]

Thank youu 🤗🤗 I want you to give an advice how to develop my skills in analysis it's seems a little bit hard isn't it?

[@josebeleno1213]

I think there's like a theorem that says that for any subset of real numbers, it's infimum is an element of the subset. Can that be proven?

[@WrathofMath]

Thanks for watching Jose! You must be mistaking that for some other theorem, or you may have just misheard/misremembered something. A set's infimum does not have to be an element of the set. For example, consider the set of rational numbers, whose squares are greater than 2. The infimum of this set is sqrt(2), which is certainly not in the set, since the set only contains rational numbers. Or consider the set of reciprocals of natural numbers { 1/n | n in N }. The infimum of that set is 0, which is not in the set.

[@josebeleno1213]

@@WrathofMath Thanks, you are right ! Things are more clear to me now.