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Title: Limit Superior and Limit Inferior
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Series: Real Analysis
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Chapter: Sequences and Limits
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YouTube-Title: Real Analysis 11 | Limit Superior and Limit Inferior
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Bright video: https://youtu.be/-y0v2V0-_8E
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Definitions in the video: Limit superior, Limit inferior, improper accumulation value
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Timestamps
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
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Subtitle in English
1 00:00:00,514 –> 00:00:03,386 Hello and welcome back to real analysis
2 00:00:03,871 –> 00:00:09,415 and as always first i want to thank all the nice people that support this channel on Steady or Paypal.
3 00:00:09,786 –> 00:00:15,031 In today’s part 11 we will talk about the limit superior and the limit inferior.
4 00:00:15,671 –> 00:00:20,700 Indeed both are very important concepts when you deal with sequences of real numbers.
5 00:00:21,300 –> 00:00:24,641 Therefore let’s immediately start with an example of such a sequence.
6 00:00:25,071 –> 00:00:29,046 Here each sequence member a_n is given by n.
7 00:00:29,743 –> 00:00:33,251 Now we know on the real number line this sequence looks very simple.
8 00:00:33,900 –> 00:00:38,934 Obviously this sequence is not convergent, because it gets as large as you want.
9 00:00:39,134 –> 00:00:45,381 So what we could say here is that this sequence is divergent, but it’s divergent to infinity.
10 00:00:46,114 –> 00:00:51,138 However please note here, we use infinity as a symbol here. Not as a number.
11 00:00:51,500 –> 00:00:59,333 More concretely by definition this should mean that for any constant I give you, we find an index capital N
12 00:00:59,943 –> 00:01:04,507 such that for all indices afterwards we are greater than this constant.
13 00:01:04,707 –> 00:01:08,927 In other words the sequence exceeds any bound eventually.
14 00:01:09,500 –> 00:01:17,825 In a similar way this definition works for a sequence that is not bounded from below, such that we can have the notion divergent to -infinity.
15 00:01:18,500 –> 00:01:21,332 There our inequalities are just reversed.
16 00:01:22,014 –> 00:01:26,861 However please note, this sequence here is not divergent to -infinity.
17 00:01:27,357 –> 00:01:32,183 Of course only one of the two properties here can occur for a given sequence.
18 00:01:32,514 –> 00:01:37,126 Another thing you should remember is that we have a symbolic notation for both cases here.
19 00:01:37,586 –> 00:01:42,382 We just write that the limit of a_n is equal to the symbol infinity
20 00:01:42,443 –> 00:01:44,809 or equal to the symbol -infinity.
21 00:01:45,300 –> 00:01:52,263 So you see, using the symbols here makes it easier to talk about the properties of a sequence that is not convergent.
22 00:01:52,463 –> 00:01:57,818 Another thing we introduced for such sequences is the notion of accumulation values.
23 00:01:58,300 –> 00:02:06,201 For this example you might have already see, this sequence does not have any accumulation value. No cluster point at all.
24 00:02:06,743 –> 00:02:13,574 However in the same symbolic way as here, we could say this sequence here clusters at infinity.
25 00:02:14,271 –> 00:02:18,447 Therefore we define the improper accumulation value infinity.
26 00:02:18,814 –> 00:02:23,236 Namely we have that for all sequences that are not bounded from above
27 00:02:23,643 –> 00:02:29,842 and of course in same way we cluster at -infinity when the sequence is not bounded from below.
28 00:02:30,557 –> 00:02:36,366 Now, this is a very nice definition, because it tells us together with the Bolzano-Weierstrass theorem
29 00:02:36,566 –> 00:02:42,154 that any sequence that as no accumulation values, has at least one improper one
30 00:02:42,586 –> 00:02:48,960 and that’s what we can use to be able to talk about the smallest or the largest accumulation value we can have.
31 00:02:49,160 –> 00:02:54,952 For this keep in mind that a given sequence a_n could have many different accumulation values.
32 00:02:55,314 –> 00:03:02,247 Of course we can visualize that on a number line as well, but now keep in mind, we could have finitely many or infinitely many
33 00:03:02,629 –> 00:03:08,719 and maybe we also have an improper accumulation value or even two outside of the number line.
34 00:03:09,414 –> 00:03:14,666 Again, infinity and -infinity are just symbols we put next to the number line.
35 00:03:15,586 –> 00:03:19,014 There are not numbers, but they are helpful for our whole description here.
36 00:03:19,786 –> 00:03:24,037 Simply because now it makes sense to talk about the largest accumulation value.
37 00:03:24,857 –> 00:03:27,879 It could be a normal one or an improper one
38 00:03:28,400 –> 00:03:30,775 and the same holds for the lowest one.
39 00:03:31,529 –> 00:03:33,984 Both of the now get very special names.
40 00:03:34,900 –> 00:03:39,178 So here we have a definition that holds for any sequence of real numbers
41 00:03:39,971 –> 00:03:48,062 and now we consider an element “a”. Either it’s a real number or the symbol -infinity or +infinity
42 00:03:48,743 –> 00:03:55,306 and now this “a” is called the limit superior of the sequence a_n, if it’s the largest accumulation value.
43 00:03:55,506 –> 00:03:59,755 So there is no other accumulation value, which is larger than “a”.
44 00:03:59,955 –> 00:04:04,400 Additionally for this we have a common notation, you might have already seen.
45 00:04:04,706 –> 00:04:08,300 We simply write “lim sup”, n to infinity of a_n.
46 00:04:09,000 –> 00:04:14,053 Ok, with this you now know one important symbol that is very often used in analysis
47 00:04:14,386 –> 00:04:19,482 and then you might not be surprised that we can do a similar thing for the limit inferior.
48 00:04:20,157 –> 00:04:25,379 It’s simply the smallest accumulation value the sequence can have or it could also be an improper one
49 00:04:26,214 –> 00:04:28,885 and then we use the notation “lim inf”.
50 00:04:29,786 –> 00:04:35,702 Ok, now you might ask why exactly do we use these strange names and these notations here?
51 00:04:36,243 –> 00:04:40,397 Therefore I would say, we invest the next minutes to discuss this.
52 00:04:40,597 –> 00:04:43,370 So let’s draw a graph for a given sequence.
53 00:04:43,914 –> 00:04:49,460 So we have the natural numbers on the x-axis and the real number line on the y-axis.
54 00:04:49,986 –> 00:04:55,687 In other words the y-coordinates of the points gives us the sequence members a_n.
55 00:04:55,887 –> 00:05:02,018 Therefore you should see at this value here and at this value, we have 2 accumulation values.
56 00:05:02,218 –> 00:05:08,084 On the other hand here you can see this point here, gives us the largest value of the whole sequence.
57 00:05:08,414 –> 00:05:14,010 In other words this is the supremum of a_k, where k goes through all the natural numbers.
58 00:05:14,886 –> 00:05:21,641 However now you might ask: what happens with the supremum, when we ignore finitely many sequence members at the beginning?
59 00:05:21,841 –> 00:05:26,265 Therefore here i just would write k is greater or equal than 1
60 00:05:26,971 –> 00:05:31,090 and then when I look at k greater or equal than 2, nothing will change.
61 00:05:31,571 –> 00:05:34,929 We still have this point here as the largest value.
62 00:05:35,129 –> 00:05:40,045 However, if we go to 3, this value is no longer considered.
63 00:05:40,245 –> 00:05:43,472 Now we find the new largest value, which is here.
64 00:05:43,672 –> 00:05:48,271 Of course it’s not possible that the supremum would be now larger than before.
65 00:05:48,471 –> 00:05:51,303 Indeed in this case it’s getting smaller.
66 00:05:51,503 –> 00:05:58,228 Moreover now it stays the same at this value, until we reach the next step, which is k is greater or equal than 9.
67 00:05:58,800 –> 00:06:06,363 This means that now we ignore all the points that are left on this side and then the largest value here is this one.
68 00:06:06,563 –> 00:06:12,819 Ok and now you might already see, we get even smaller when we increase the number here even more.
69 00:06:13,286 –> 00:06:17,291 For example we can choose 11 and then we find this value here
70 00:06:17,700 –> 00:06:20,694 and then the supremum stays at this level.
71 00:06:20,894 –> 00:06:23,281 No matter how big this number here is
72 00:06:23,481 –> 00:06:29,337 and please recall, we already know this value here is the largest accumulation value.
73 00:06:29,537 –> 00:06:33,016 Therefore by definition, this is the limit superior
74 00:06:33,471 –> 00:06:39,200 or in summary we now have learned that the limit superior describes what happens with the supremum,
75 00:06:39,257 –> 00:06:43,100 when we cut of more and more at the beginning of the sequence.
76 00:06:43,800 –> 00:06:46,297 In this sense we have the following fact.
77 00:06:46,757 –> 00:06:54,942 The lim sup of a sequence is related to the sequence given by the supremum, where n the index is the cutoff.
78 00:06:55,142 –> 00:06:58,418 So we have k is greater or equal than n.
79 00:06:58,729 –> 00:07:02,250 So this is a well defined sequence with index n.
80 00:07:02,450 –> 00:07:07,979 The only strange thing that can happen here, is that all the sequence members here are the symbol infinity.
81 00:07:08,386 –> 00:07:13,052 However with the exception of this case, we get a well defined sequence of real numbers,
82 00:07:13,252 –> 00:07:17,145 where we already know, this sequence is monotonically decreasing
83 00:07:17,671 –> 00:07:22,147 and then we look at the limit of this sequence, when n goes to infinity
84 00:07:22,729 –> 00:07:29,197 and now one can show, this is exactly the largest accumulation value. Therefore our limit superior.
85 00:07:29,729 –> 00:07:36,566 Here please keep in mind, on both sides infinity and -infinity as symbols are possible.
86 00:07:36,766 –> 00:07:41,883 Also i can tell you, often this one is used for the definition of the limit superior.
87 00:07:42,514 –> 00:07:47,563 However still there it’s a good exercise to show the relation between the accumulation values there.
88 00:07:48,371 –> 00:07:52,885 Ok, maybe not so surprising, we have a similar result for the limit inferior.
89 00:07:53,343 –> 00:07:59,456 There we just observe what happens with the infimum, when we cut of finitely many sequence members at the beginning
90 00:08:00,000 –> 00:08:04,854 and what we get is a sequence with index n, which is monotonically increasing.
91 00:08:05,400 –> 00:08:11,629 However also here -infinity and infinity as symbols are possible on both sides.
92 00:08:11,943 –> 00:08:19,485 Ok, then I would say, examples and more properties of the limit superior and limit inferior we can discuss in the next video.
93 00:08:19,886 –> 00:08:23,314 Therefore I hope I see you there and have a nice day. Bye!
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Quiz Content
Q1: Consider the sequence $(a_n){n\in \mathbb{N}}$ given by $$a_n = (5,-5,1,2,1,2,1,2,1,2,1,2,\ldots)$$ What is the limit superior of $(a_n){n \in \mathbb{N}}$?
A1: 5
A2: -5
A3: 1
A4: 2
A5: 0
Q2: Consider the sequence $(a_n){n\in \mathbb{N}}$ given by $$a_n = (5,-5,1,2,1,2,1,2,1,2,1,2,\ldots)$$ What is the limit inferior of $(a_n){n \in \mathbb{N}}$?
A1: 5
A2: -5
A3: 1
A4: 2
A5: 0
Q3: Which one of the following statements is not possible for any sequence $(a_n)_{n\in \mathbb{N}}$ consisting of real numbers?
A1: $ \displaystyle \limsup_{n \rightarrow \infty} a_n = -\infty$
A2: $\displaystyle \liminf_{n \rightarrow \infty} a_n = \infty$
A3: $ \displaystyle \liminf_{n \rightarrow \infty} a_n = \limsup_{n \rightarrow \infty} a_n$
A4: $\displaystyle \liminf_{n \rightarrow \infty} a_n \neq \limsup_{n \rightarrow \infty} a_n$
A5: $ \displaystyle \liminf_{n \rightarrow \infty} a_n > \limsup_{n \rightarrow \infty} a_n$
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Last update: 2025-01
