What is Limsup ?
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- Опубликовано: 9 ноя 2024
- 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
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I like how Dr Peyam "ignored" the past. Seriously, with Dr Peyam on math is like watching a movie.
Blessed with a Dr. Peyam video beyond where I got in analysis. Gonna be a good day!
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!
Thank you!!!
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
Thanks so much!!!
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!
Thank you!
Dr Peyam, I love how you teach mathematics. Thats exactly how i tutor or at least hope to tutor my students.
Dear Dr. Peyam, you're amazing.
Thank you!!
I haven't even gone far into the video and I can say this is amazing. Thank you sir.
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.
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.
Hahahaha interesting
limsup notation cause lots of confusion when I was learning Rudin. Thank you. This one is very intuitive.
Oh my, this is so easy that the question I don’t know is why I could not understand it so far???
It’s actually a pretty hard topic, I just explained it well 😊
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
Also of interest is that the limsup and liminf are the largest and smallest values that are limits of some subsequence. (Such limits can be proved to exist)
This is Gold. Knowledge is the real Gold
I was waiting for a better explanation on liminf and limsup. I got it here. Thanks Dr. Peyam..
We want the Bolzano weistrass theorem proof dr peyam
It’s in the playlist
@@drpeyam no dr peyam I mean the Bolzano weistrass theorem in R^n that says that an infinite points on a compact set has at least one limit point
@@josephhajj1570
I think the theorem demonstrates the existence of limit points in infinite bounded sets, not necessarily compact.
In that case, I don’t know the proof either, but observe that infinity is not a sufficient condition: Z is infinite, although it doesn’t have any limit points. Hence, it seems that being bounded is another necessary condition.
@@josephhajj1570 you can lead that one back to the one dimentional. Just apply it to every part of the sequence vector :) there u go :)
thank your, professor, really simple explanation in such few words and drawings, i finnaly understand the definition now =D
Yay!!!
for the very first time I do understand this thing. Thank you very much!!!!!!!
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!
Great video, very clear! Great prep for my PhD in maths
Thank you!!
Thankyou sir. I was really scared of this topic. Thanks for erasing my fear :)
All I was able to hear after approximately 9:30 is lime soup :).
thank you so much, i don;t understand lim sup and inf until now.
You are a hero Dr Peyam
The lim in the limsup scared me (also the sup scared me), but the idea is straightforward.
We define the limsup of a sequence s_1, s_2, ... , s_n, ... , as the limit of a sequence of sups, specifically,
sup {s_1, s_2, ... }, sup {s_2, s_3, ...} , ..., sup { s_3, s_4, ... } , .. (this is made rigorous in the youtube video).
The threshold-wall analogy is great. We can visualize the limsup as the sup of the graph of (n, s(n)) as you keep moving the wall to the right (so you cannot see initial terms to the left). Alternatively, fix the screen and move the points (n, s(n)) to the left, and take the sup of the terms remaining to the right (like a mario brothers game ).
Moreover, at the risk of being handwavy, limsup n→∞ s_n = sup { s _∞ , s_∞ + 1, ... } .
Note that the latter is only meant to be a heuristic, not a rigorous statement.
Also note for the ultrafinitists, the symbol ∞ is intended to be merely suggestive of a very large number (in this context a very large integer) . This permits us to avoid the whole 'potential infinity' versus 'completed infinity' debate.
Most handsome and talented mathematician
*Cutest also 😍😍
By far the happiest
Greatest explainer, I subscribe people rarely, here have a subscriber
Thanks so much!!
Very helpful video,concept cleared 😇,
Thank you. Simple and clear
Thank you sir for this super video so clearly explaining the concepts 9f limsup and living !
Man you are a life saver
Thanks so much Dr
Well after watching the video, I reckon that limsup should be some delicious soup.
best explanation possible
This is the best explanation ever!!!!!!!
thanks for explaining this so clearly - it helped me.
Great video! There seems to be one more option though besides the limit superior going to infinity or converging to a finite number: It could be negative infinity also.
Clear. Thank you very much.
Thank you so much , great explanation
excellent explanation, thank you
Will you make this video public soon?
Yeah
dr peyam, essential supremum next?
Thanks, great explanation! Btw, Why is V_n bounded below? You assumed it, right?
I believe the sequence is bounded
@@drpeyam Got it. Thanks.
Hey Dr Peyam.. Could you please do a video which talks about lim sups and lim infs as the greatest and the least subsequential limits of a sequence? I understand both the definitions of lim sup separately (i.e 1) as the greatest subsequential limit and 2) in terms of helper sequences) but can't quite understand why these two characterizations are equal. :-/
I’ve already done that, check out the playlist. Something like subsequence converging to Limsup
@@drpeyam Yayy!! Awesome
Dr peyam, if a sequence is bounded, will it always be that the sup is larger before the lim sup before N? Is this why V is a decreasing sequence? What if it started off small before it started oscillating? The analogy if the class grades wouldn't apply then?
I get it now actually, even if the sequence starts off small, its the supremum of all the terms from x n onwards, I get it
There is an alternate definition of lim sup: lim sup is defined is as the largest limit point (or accumulation point) of the sequence. Let's call this def 2 and yours def 1.
I can prove def 1 implies def 2. But I am unable to prove def 2 implies def 1. Please help. Any relevant link is suffice. Thank you!
That was really really helpful thank you!
what if the you have something like 1/x which has a pole at zero. if you define Sn=1/(n-10) for example at 10 it diverges but after that it goes down, so it's not bounded, but when n is large it's going to tend to 0. ?
That’s why we write n goes to infinity
S_10 isn't defined here? Do you really have a sequence then?
after 2 year of my college now I come to know that ooh limsup was this......
Makes perfect sense now
This is amazing, thank you!!!
Wow! Thank you!
10:16
You say supremum always exists. So, even for the sequence = N (natural numbers), would sup be infinity? Is it ok to say sup = infinity?
Yep! There’s actually a video on that in the playlist
Like R barre
You've gained a subscriber
Awwwww thank you!!!
Thanks
Bolzano-Weierstrass is the next step?
It’s in the playlist
Sir, where are u from?
I wish I could give multiple likes to this video!!!♥️♥️😘😘
can you explain why the limsup is still decreasing when the sequence (for example, at 2:30) is monotonically increasing and is bounded above? I don't quite get that part when reading the textbook. I get the idea of the proof of why {a_n} (defined as sup{x_k : k greater equal to n}) is decreasing and {b_n} is increasing, but visually, what if the sequence is monotonically increasing and bounded above? Isn't the sup also increasing in that case? What is the inf? Thank you so much!
In the case of an increasing sequence the Sup would just be constant every time so it’s increasing in a degenerate case. The inf would be the value at N
@@drpeyam Thank you so much for the reply!
Mathematicians are always coming up with new definitions to sort out the singularities that former definitions cannot fix. Like cesaro sums, analytic continuity, lim sup and so on.
Thaks bro thats realy help
Lin-soup is the soup that in the long run !!! That is the whole point.
Wow! Thanks.
What happens to liminf{x_n} if {x_n} is only bounded above? Does liminf{x_n} differ depending on how the sequence {x_n} looks like?
It could be -oo as the example xn = -n shows
What if you had a limsup such that it fit a function rather than a constant. Perhaps you could expand models of limsup with lim arbitrary function f(x)
Just take limsup at every point. It’s called the upper semicontinuous envelope I believe
What if a sequence was to increase and then oscillate, and the limsup is larger than any other element before N, does that mean that V(N) is still decreasing?
It’s still decreasing, since the larger N is, there are fewer values to compare to. The sup of 20 elements is always bigger than the sup of a subset with 10 elements
Dr Peyam Oh I see, thank you!
I enjoyed the amount of soup in this one =)
Good soup 🍜
So if limsup = liminf, does that imply lim converges?
Exactly, there’s a video on that
If you have a decreasing unbounded sequence, would limsup be -inf?
Yes since the limit is -inf
Awesome kept it up!
This is god’s work 👏🏼
One question Professor:
Is limsup at infinity is the same as 'sup infinity'?
I never heard of the latter
Now, I'm no mathmatical wizard and my formal education in math is limited to the "need to know"-level of economists. Now my math professor in university was one of the few I bothered to listen to.
But I do find the level of ignorance and interest disconcerting. I know the abysmal lack of comprehension for a fact. My farther did a course in pharmaco-kinetics shortly before retiring and I as a student was able to derive the fairly simple first order differential equations.
The elimination of a drug is a straight line on single logarithmic paper.
The problem were that neither his collegues nor the medical professor GOT IT! It simply flew into interstellar space over their pin-heads.
There was a nobel laureate in economic that had a model of the price of shares based on the assumption that stock market quotes were normal distributed - probably because it was the only statistical distribution he knew. The result was as devastating as it was predictable: He ruined his business because in so far as the stock market quotes are statistically distributed they are NOT following a normal distribution. The Normal Distribution is an extreemly "slimtailed" distribution which does not take rare but very extreme occurences into consideration.
Personally I have given up - I do not want to waste my life telling people that they are ignorant idiots, as they will never believe it - no matter what. So I have resigned myself to be the nasty disrespectfull person. There are no perks in being right.
It's a simple fix really. Just get your own Nobel prize because you can easily beat the chump who doesn't understand normal distributions and with the money open a more successful business which will never collapse in your lifetime. Only perks in being right are when you apply it to your own decisions.
Very cool!
What is MCT? Is it Monotonic Convergence theorem?
Yeah
@@drpeyam thank you Dr! Can you explain about how to use completeness axiom in finding the sup and inf of an interval? This analysis course is really struggling for me 😢
Check out the playlist on Real Numbers
@@drpeyam noted with thanks Dr!
Love you
"Sequence Vn is not always decreasing, but not increasing". But what about arctg(n)?
vn is non-increasing, but it could be constant
@@drpeyam But the function arctan (x) is increasing
No but look at the definition of vn
@@drpeyam Thank you, I understand, I saw the picture comparison values and the definition I overlooked.There are Suprems , not values.
Yes
Bro I love u hahahah best explanation ever
Thank you ❤️
Is similar things valid for liminf or not?
Of course
If S n is bounded
Sup{S n | n> N} = 1
For all N
Sup biggest value in sequence
Lim sup Sn lim [Inf]= 1
Limsup[Inf] = sup (biggest value) converging in the long run
Has to do with the helpers sequence
Sequence decrease
& non increasing
Sometimes limit
But the bright side
Limsup[inf] always exists
Lim Sup doesnt exist
Infimum the smallest value
Lim inf [Inf] {SN | n >N}
Fact lim inf sequence fn Is
Lim inf [Inf]=-lim sup [Inf] -Sn
Recall inf S = - sup -S
Long run
Inf {Sn | n > N }=
- sup{ - Sn | n >N }
Taking lim of both sides
Lim inf Sn[Inf] = - lim sup Sn[Inf]
Sup?
limsoup
A soup made from human limbs. Scary
These ideas aren’t so difficult, it’s just that the most textbooks make a poor job at explaining them
Was it this simple...
why did you start at Vsubscriprit("0") , 0 is not a natural number
It is
Wa lmo3a9
not much, what's limsup with you
Most other videos and time wasting on other channels
Not much what about you
is he high on math?
limsup balls lmao gottem
thanks