Proof: Maximum of a Set is the Supremum | Real Analysis

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The maximum of a set is also the supremum 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 maximum value m, then the maximum m is also the supremum of the function.

Recall that the maximum of a set A is an element m in A such that m is greater than or equal to every element a of A. The supremum of a set A is the least upper bound: as in, of all values that are greater than or equal to every element of A, the least such value is the supremum, written as sup A. Not every set has a maximum or a supremum, but if a set has a maximum then that max is also the sup!

I hope you find this video helpful, and be sure to ask any questions down in the comments!

+WRATH OF MATH+


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Комментарии
Автор

Could you take proving questions on functions ?

diyasehgal
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All is forgiven. The proof is succinct and understandable. Thanks!

rmw
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but it's possible to have a supremum/infimum that is not a max/min, correct?

jacksonmadison
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The problem I have with most YouTube teachers just like you is the rush when explaining. If someone is watching this video is a proof he/she has real difficulty in understanding this topic, so why the

maxmillian
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Sorry i want to say sup( A intersects B ) <= max ( sup(A), sup (B)), yeah a proof please, if that’s possible
, thanks for your efforts

anonymousvevo
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bro, do you have a video of the following demo
A intersects B <= MAX(sup(A), sup(B))
thank you

anonymousvevo
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why is m' not greater or equal than a?

froglet
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Hey, can you please prove this in your next video as soon as possible:
Let G be a simple graph without an even cycle. Prove that the graph G has an independent set of
size at least n/3.

PLEASE!! Thank you so much!!

shivangitomar
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We have to show sup(S) =max(s)
So for contradiction sup(s) /=max(s)

adnansohail