Real Analysis 11 | Limit Superior and Limit Inferior

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This is my video series about Real Analysis. We talk about sequences, series, continuous functions, differentiable functions, and integral. I hope that it will help everyone who wants to learn about it.

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00:00 Intro
00:20 Example
02:07 Improper accumulation value
03:34 Definition limit superior and limit inferior
04:29 Why do we use these names and notations?
06:45 Fact
08:24 Credits

#RealAnalysis
#Mathematics
#Calculus
#LearnMath
#Integrals
#Derivatives

I hope that this helps students, pupils and others. Have fun!

(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[lala]

thank you SO much. You saved my PhD in economics

[kevinhu]

Not all heroes wear cape. And you became my hero right on time.

[lidiias]

Very underrated channel. Thanks for such a clear explanation!

[mathsandsciencechannel]

Wow. Love this sir. I love solving challenging calculus, algebra, trigonometry and geometry maths problems on my.... Thanks

[douglasstrother]

The accumulation value of the sequence "Free Beer Tomorrow" is infinity.

[rayx.]

I have been learning lots of useful things from your channel! Many thanks! Could you make a video on limit superior and limit inferior of sequence of sets (measure theory)? I found it very hard to clearly grasp and so far haven't found good videos on it. Tons of thanks in advance!

[awesomecraftstudio]

Getting math explained to me by The Bright Side of Mathematics >>> My crush telling me she likes me

[qiaohuizhou]

2:07 improper accumulation point
2:35 every sequence has at least one improper accumulation value
3:35 limit sup and limit inf
4:42 why intro the concept
6:45 fact

[anubhavsingh]

Thanks a saved my engineering degree at IITD

[maxpercer]

1:05 In my humble understanding, this is not the definition of 'divergence of a sequence'. It is a special case of 'divergence of a squence' which applies to sequences that eventually blow up. Sequences can also oscillate. For example the values in the sequence { 1, -1, 1, -1, ... } do not eventually exceed all values C > 0, e.g. C = 2, but the sequence is still divergent. There are other more interesting oscillating examples .

[daviidayala]

This chapter reminds me of an exam question that was "proove that there exists a sequence of numbers where for any real value a, a is the limit of a subquence of the original sequence". The answer was very obvious though no one thought about it during the exam :p

[froglet]

this concept is useful in measure theory, for sequences of elements of sigma algebras

[sarthakgupta]

At 1:30 you say that, only one of the two properties here can occur for a given sequence. What about (-1)^n * n?
The terms get arbitrarily large with alternate sign.

[scollyer.tuition]

Nice graphical demonstration of lim sup. And I commend you for the fact that you put arrows pointing +ve on your axes - it drives me crazy when people omit them.

One trivial point though: you seem to be writing a single word "limsup" which looks a bit odd to me. Latex (at least) formats \limsup with a small space between lim and sup. Not sure if that is standard or not.

[sarthakgupta]

I am a bit confused. What do you mean by the largest accumulation value? For example, the largest accumulation value amongst 2, 37, 14, 6985, 23 and 5 is 6985, right? If I am correct then, the limit supremum at 6:40 should be a_2, not a_12. Please help me.

[JoopWilkens]

Why do you use the terminology 'divergent' to \infty? Isn't it more natural to call it *convergent* to \infty?

[tomibozak]

1:28 what about (-2)^n? It is not convergent, and it goes to +-infinity. so it is not divergent either?

[lucasm]

What about a_n = (-1)^n * n? It’s approaching both infinities

[pirzadaaakib]

Sir, I have a question. Do finite sequences converge? Please answer it with some examples

[Zumerjud]

Doesn't the Bolzano-Weirstrass theorem apply to bounded sequences?