Glad to help! I plan on doing a lot of work to build up the playlist this summer, hopefully it will continue being a useful substitute for years to come!
Your explanation is really helpful! Thanks again for taking my request, you're like a mathematical DJ :) Just curious, where are these exercises from? Also, your sweater is excellent 🐸
@Wrath of Math I have looked in the description for a link to the videos with the proofs but I couldn't see any. I would love to watch them if you can kindly point out where I can can find them. I love your videos please keep them coming :)
this was a great explanation, thank you. but i'm still stuck on how to work with limsup/liminf if it tends to something else other than infinity (e.g. 0 or 1) since I don't know where you would start 'cutting off' terms.
for example, i wanted to check whether the limit exists for a_n = lim_(x -> 1) (cos(1/lnx)). I calculated limsup and liminf as x goes to infinity and they were both 1. does this mean that the limit exists for a_n? Or do i have to calculate limsup and liminf as x goes to 1 instead?
A dedicated explanation as usual. But I wonder, so can it be the case that: a sequence has all of its terms be less than (or =) some constant C, but C may not be limsup(seq). It can be a graph where in the first N-th terms, it increases to a very value C . After that, it decreases to some value and then converge to some number D that's lower than C. Is my interpretation valid? I personally think we should include these confusing examples to strenghthen the understading of the material.
Thanks for the question! If I understand you right, the answer is no. For example, maybe our sequence seems to approach C for the first 100 terms and then starts decreasing to some number D. Then if we consider the Supremum of the terms after term 101, that Supremum will necessarily be less than C because the 101th term, which we assume has decreased from C, is the Supremum of the set of terms succeeding the 100th term. By similar logic, we can see the limsup will not be C because the Supremum of the tails will continue to decrease as we go further in the sequence.
@@WrathofMathSorry for my lack of clarity. I mean that the sequence increases to C for the first 100 terms, then decreases. But: only for a FINITE number of steps, after that, the sequence increases again while also converging to D. And we assume D
Very clear explanation. Even for beginners. I think it is much clearer than the one from 'The Bright Side of Mathematics'.
Thank you! Hopefully between Bright Side and myself we can offer explanations for all real analysis topics people will be able to understand well!
thanks man, your explanations are crystal clear and are really helping me for my grad school preparations.
It’s my pleasure - thanks for watching and good luck with grad school!
hello great video i was wondering where the proofs are that you mentioned you would put in the description thank you again
This is great!!!! I was having a hard time understanding the defn from the textbook, turns out it only takes one search from yt to get this over with.
Glad to help - thanks for watching!
Great video - thanks very much - your real analysis playlist has become my substitute for lacklustre lecturers.
Glad to help! I plan on doing a lot of work to build up the playlist this summer, hopefully it will continue being a useful substitute for years to come!
You are really a wizard.
Thank you!
You the man🙏🏿 I hope you have a great day
Thanks a lot Nceba, same to you!
Your explanation is really helpful! Thanks again for taking my request, you're like a mathematical DJ :)
Just curious, where are these exercises from? Also, your sweater is excellent 🐸
So glad it was helpful! That's the goal, haha! I used to turn lesson requests around in
Thank you
You're welcome!
Wonderful Video,sir😊🙏🌠
Thank you!
Really really wonderful 😊
Thank you! Cheers!
Thank you so much this was so helpful
Glad to hear it, thanks for watching!
Let's get to differentiation my man!
I am going as fast as I can!
Taking RA rn- this video helped a lot thanks! (Subbed)
@Wrath of Math I have looked in the description for a link to the videos with the proofs but I couldn't see any. I would love to watch them if you can kindly point out where I can can find them. I love your videos please keep them coming :)
Those videos are not done yet, I will add their links to the description as soon as they are! Thank you for watching and sorry for the delay on those!
Thank you !!!
Glad to help!
Muy buen video crak
this was a great explanation, thank you. but i'm still stuck on how to work with limsup/liminf if it tends to something else other than infinity (e.g. 0 or 1) since I don't know where you would start 'cutting off' terms.
for example, i wanted to check whether the limit exists for a_n = lim_(x -> 1) (cos(1/lnx)). I calculated limsup and liminf as x goes to infinity and they were both 1. does this mean that the limit exists for a_n? Or do i have to calculate limsup and liminf as x goes to 1 instead?
Is it ok to write for limes inferior=5 and limes superior =7 as an=5+2/n
A dedicated explanation as usual. But I wonder, so can it be the case that: a sequence has all of its terms be less than (or =) some constant C, but C may not be limsup(seq).
It can be a graph where in the first N-th terms, it increases to a very value C . After that, it decreases to some value and then converge to some number D that's lower than C.
Is my interpretation valid?
I personally think we should include these confusing examples to strenghthen the understading of the material.
Thanks for the question! If I understand you right, the answer is no. For example, maybe our sequence seems to approach C for the first 100 terms and then starts decreasing to some number D. Then if we consider the Supremum of the terms after term 101, that Supremum will necessarily be less than C because the 101th term, which we assume has decreased from C, is the Supremum of the set of terms succeeding the 100th term. By similar logic, we can see the limsup will not be C because the Supremum of the tails will continue to decrease as we go further in the sequence.
@@WrathofMathSorry for my lack of clarity. I mean that the sequence increases to C for the first 100 terms, then decreases. But: only for a FINITE number of steps, after that, the sequence increases again while also converging to D. And we assume D
In that case D would be the supremum, since it is the supremum of the tail, and limit supremums concern the "tail" behavior