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

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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.

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

a.robberts

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?

kaoutherlhb

Super helpful! More real analysis videos please!

catherineferris

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?

josebeleno

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