What is Limsup ?

Definition of Limsup

In this video, I define the single, most important concept in analysis, the Limsup of a sequence. It is like a limit, except that it always exists. Enjoy this video that is fundamental to your analysis knowledge

[HarpreetSingh-kezk]

I like how Dr Peyam "ignored" the past. Seriously, with Dr Peyam on math is like watching a movie.

[KhanhNguyen-jqni]

I'm learning real analysis for a short course in optimization, and this is the clearest explanation of lim sup that I've ever witnessed. You rock Dr Peyam!

[bertrandspuzzle]

Blessed with a Dr. Peyam video beyond where I got in analysis. Gonna be a good day!

[lmbcompany]

You helped me think a little wiser about the limits and more... I don't subscribe to people easily but I did it in you just because of your love about maths(simple or more complicated, doesn't matter). Keep going. Hard work beats tallent!

[Mr_mechEngineer]

Dr Peyam, I love how you teach mathematics. Thats exactly how i tutor or at least hope to tutor my students.

[IamRigour]

I haven't even gone far into the video and I can say this is amazing. Thank you sir.

[takoeatsall]

My god you are a lifesaver. I came to grads school of statistics without proper math skills, and couldnt understand my professor going over limsup and liminf on day 1. You are perfect teacher

[iabervon]

I think the interesting sequences are ones like s_n = (1+1/n)cos n, where for all N, there's an n > N such that s_n > limsup s_n and s_n also doesn't converge. And limsup s_n isn't a term of the sequence, either.

[uniedits]

thank your, professor, really simple explanation in such few words and drawings, i finnaly understand the definition now =D

[laralahey]

An excellent video, I have been struggling with understanding the intuition behind this concept; but you have answered all of my questions. I can't thank you enough!

[amberheard]

limsup notation cause lots of confusion when I was learning Rudin. Thank you. This one is very intuitive.

[davidkwon]

Oh my, this is so easy that the question I don’t know is why I could not understand it so far???

[Songvbm]

I was waiting for a better explanation on liminf and limsup. I got it here. Thanks Dr. Peyam..

[felipegomabrockmann]

for the very first time I do understand this thing. Thank you very

[FT]

My analysis prof constantly told us to "play with the soup" to prove things, etc. I can only ever think of lime soup.

Also, we defined it as the inf of all the sups, rather than a limit of the sups. Seems to be the same thing.

Do you have to mention the case where the sequence isn't bounded from below? Then the limsup would be -infinity.

[mtaur]

To fully grasp the concept, realize that the sequence may exceed its limsup infinitely often, but only by amounts that approach zero. Such as s_n=(1+1/n)(-1)^n

[rupeshmishra-vnc]

Most handsome and talented mathematician

[josephhajj]

We want the Bolzano weistrass theorem proof dr peyam

[frozenmoon]

All I was able to hear after approximately 9:30 is lime soup :).

[harrisonbennett]

Great video, very clear! Great prep for my PhD in maths