Proof: Supremum and Infimum are Unique | Real Analysis

If a subset of the real numbers has a supremum or infimum, then they are unique! Uniqueness is a tremendously important property, so although it is almost complete trivial as far as difficulty goes in this case, we would be ill-advised to not prove these properties! In this lesson we'll be proving the uniqueness of suprema and infima.
Recall that the supremum of A, denoted sup A, is the least upper bound of A. The infimum of A, denoted inf A, is the greatest lower bound of A.
Definition of supremum and infimum of a set: • Definition of Supremum...
Epsilon Definition of Supremum and Infimum: • Epsilon Definition of ...
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I hope you find this video helpful, and be sure to ask any questions down in the comments!
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[@Zinani-z7b]

Super interesting proof. I didn't expect it to be so straightforward. Thank you very much. Have a good day! ❤️

[@WrathofMath]

Glad to help - thanks for watching!

[@mirandaatangdithebe3893]

great video, but could you do a proof showing that a infimum of a set exists while the min does not

[@anjaneyasharma322]

Blue M In such functions value of f(x) keeps decreasing an approx value is taken and called infimum. The sum of the series is found for as x tends to infinity.

[@mahmoudalbahar1641]

Many thanks for this good video.

[@valeriereid2337]

This for this explanation. It most certainly helped.

[@siphendulwezaza9384]

Can Supremum and Infimum of a set be the same?

[@WrathofMath]

Great question, they absolutely can! Consider { 1 }. For this set, 1 is the least upper bound and the greatest lower bound.

[@siphendulwezaza9384]

@@WrathofMath Thank you

[@anjaneyasharma322]

A line parallel to X axis

[@ririsantosratunil9514]

Thank you!!!

[@WrathofMath]

No problem! Thanks for watching and let me know if you have any video requests!

[@ahmad_qau]

Thank you. It was quite useful. 👏👏

[@WrathofMath]

Glad it was helpful!

[@lalhriatpuiahmar5057]

How can there be a two different suprema s1 and s2 if it is unique. Is it considered to be both s1

[@WrathofMath]

Thanks for watching and good question! There aren't two different suprema, which is what we prove! We say s1 and s2 are both suprema. We give them different names so that we don't implicitly assume they are equal, but then we prove they must be equal by definition, and so there is only one supremum!

[@realndu]

Thanks. Damn. You make these seem easy