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

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!

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$