Supremum and infimum EXAMPLES - Part 1 - Real ANALYSIS - Mathematics

Gain an intuitive understanding of supremum and learn how to prove that a number is the supremum of a set. Understand why the concept of supremum is important and why \sqrt(2) exists as a real number. Review the Completeness Axiom of the real numbers.
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[@helgabigear1255]

Great job, Big Epsilon! I'm a Russian teacher of math who worked for 20 years at the technical university. I wish I can speak English so fine... Despite of my natural hearing disabilities, I can hear every word you say clearly. Your voice is beautiful. The examples you eplained aren't complicated, but the way you are teaching is brilliant. Waiting for more videos from you. Thanks a lot!

[@BigEpsilon]

So glad you found my videos helpful :) Good wishes and welcome to the channel

[@sandralettu1631]

I love your videos so much. Thanks for making me enjoy discrete maths.

[@BigEpsilon]

You are welcome!

[@xoppa09]

the sup of a set is the best upper bound

[@kaos092]

The thing that confuses me about Supremum and infimum is that in sets that don't include the number such as (-infinity, 2) then 2 is the supremum. But 2 isn't part of the set. Does this mean that supremum and infimum are like limits in that you just get as tightly bounded to the set as you can from either side?

[@BigEpsilon]

You got it. For the supremum, there exists a sequence of elements within the set that converges to the supremum. If the supremum belongs to the set, it is called a maximum. If the set has a maximum, that maximum is the supremum. But some sets like (-\infty,2) do not have a maximum: 1.9 is not a maximum because 1.99>1.9 is in the set, and 1.99 is not the maximum because 1.999>1.99 is in the set, and so on. So (-\infty,2) does not have a maximum, but it does have a supremum.

[@totallyalaa]

great one helped me a lot thanks ! can you give me a book name please that contain this topics

[@BigEpsilon]

Glad you found the video helpful! There are many books on this topic, look for books on "modern analysis" or "real analysis", this is a standard topic. I like the book by Wade.

[@sumittete2804]

Ma'am...what are the supremum and infimum of (0,1) intersection {m+n√2 : m,n are integers}?

[@BigEpsilon]

Is this your homework question? I would plug in a few integer values of m,n into m+n\sqrt(2) to see what numbers you get, just to get an idea so you can make an educated guess.

[@koriware9231]

I dont get how it can be 2 when if we come back to the beginning question it says r^2

[@BigEpsilon]

Can you clarify what you mean by "it can be 2"?

[@koriware9231]

@@BigEpsilon your saying the limit is 2. but the original question is: r^2

[@BigEpsilon]

@@koriware9231 Thank you for your question, Kori. There is no talk about limits in this video, it is about supremum. The example you are talking about states that the set E={r \in Q: r^2

[@koriware9231]

@@BigEpsilon thank you. Yes that was a very hard concept for me to grasp. I thought the supremum was a limit. No wonder I was so lost.

[@BigEpsilon]

@@koriware9231 No worries, this is a very common misconception, glad I could clarify