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Title: Subsequences and Accumulation Values
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Series: Real Analysis
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Chapter: Sequences and Limits
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YouTube-Title: Real Analysis 9 | Subsequences and Accumulation Values
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Bright video: https://youtu.be/xZ5vjdZzTUI
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Dark video: https://youtu.be/TRTA66eXd-8
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Ad-free video: Watch Vimeo video
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Exercise Download PDF sheets
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: ra09_sub_eng.srt
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Definitions in the video: subsequence
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Timestamps
00:00 Intro
00:13 Definition of subsequences
01:40 Example
03:26 General fact about subsequences
04:00 Example with a divergent sequence
05:04 Definition accumulation value
06:49 Alternative definition of accumulation values
07:47 Credits
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Subtitle in English
1 00:00:00,371 –> 00:00:03,241 Hello and welcome back to real analysis
2 00:00:03,971 –> 00:00:08,800 and as always many many thanks to all the nice people that support me on Steady or Paypal.
3 00:00:09,400 –> 00:00:13,957 Today in part 9 we will talk about subsequences and accumulation values.
4 00:00:14,486 –> 00:00:19,214 For this let’s immediately start with a picture of a sequence on the real number line.
5 00:00:19,786 –> 00:00:24,279 There we have a lot of points called a_1, a_2, a_3 and so on.
6 00:00:24,786 –> 00:00:30,748 Now, the term subsequence is not so hard to understand. We just omit some members of this sequence.
7 00:00:31,114 –> 00:00:39,651 For example we can take the 1st one, but then omit the 2nd one, but then again take the 3rd one and 4th one and then omit some other ones.
8 00:00:40,214 –> 00:00:44,265 The only thing we need is that we still have infinitely many members.
9 00:00:44,971 –> 00:00:47,457 So we still have a sequence in the end.
10 00:00:47,971 –> 00:00:51,137 However then we need a new name for the sequence members,
11 00:00:51,337 –> 00:00:55,964 because a sequence is a map from the natural numbers into the real numbers.
12 00:00:56,343 –> 00:01:02,288 Therefore the first member needs the index 1 and the second member the index 2.
13 00:01:03,086 –> 00:01:07,652 Therefore usually we use n as an index with an additional index.
14 00:01:08,229 –> 00:01:13,643 Hence here you should see, n with index k is a sequence of natural numbers.
15 00:01:14,100 –> 00:01:17,758 Moreover it needs to be strictly monotonically increasing.
16 00:01:18,586 –> 00:01:23,586 This simply means that for all sequence members the successor gets strictly greater.
17 00:01:24,471 –> 00:01:31,184 Then in this case the sequence with index k here, is called a subsequence of the original sequence a_n.
18 00:01:31,757 –> 00:01:39,773 So this is our first definition for today. We are allowed to omit some members, but then we need a new enumeration given by k.
19 00:01:39,973 –> 00:01:43,490 Ok, maybe it gets clear when we look at an example.
20 00:01:43,971 –> 00:01:46,657 For this lets take a sequence we already know.
21 00:01:47,057 –> 00:01:50,851 Namely a_n is given by 1/n.
22 00:01:51,671 –> 00:01:57,699 Now the question is: what is the subsequence, when we choose n_k as 2 to the power k?
23 00:01:57,986 –> 00:02:01,171 Please note this is a possible choice for the sequence n_k,
24 00:02:01,257 –> 00:02:07,100 because 2 to the power k is a sequence of natural numbers, that is strictly monotonically increasing.
25 00:02:08,014 –> 00:02:15,071 Now, this means in the original sequence here, we will omit all the members that have an index that is not a power of 2.
26 00:02:15,929 –> 00:02:22,501 Therefore the first one should be 1/2, the next one 1/4, then we go to 1/8 and so on.
27 00:02:23,071 –> 00:02:26,782 Hence our subsequence a_n_k looks like this.
28 00:02:27,457 –> 00:02:33,313 One important thing you should note here, is that we can’t change the overall order the sequence had before.
29 00:02:33,814 –> 00:02:40,658 This means that in the case that the sequence visits 1/2 only once and also 1/4 only once
30 00:02:41,286 –> 00:02:51,364 and when know in addition that 1/4 comes after 1/2, then it’s not possible that in the subsequence we have 1/4 before 1/2.
31 00:02:52,129 –> 00:03:00,240 In particular if we have such a monotonic sequence, we already know that the subsequence is also monotonic in the same way.
32 00:03:00,771 –> 00:03:06,761 Another thing you should note here is that we can omit a lot of sequence members, even infinitely many.
33 00:03:06,961 –> 00:03:10,123 As long as infinitely many remain.
34 00:03:10,323 –> 00:03:15,995 Ok, one thing we didn’t discuss yet, is what happens with the limit if we go to a subsequence.
35 00:03:16,729 –> 00:03:22,637 In our example here you see, the limit is 0 for this sequence, but also for this sequence.
36 00:03:23,157 –> 00:03:25,984 Or in other words we don’t change the limit here.
37 00:03:26,329 –> 00:03:29,783 Indeed this is a general fact that always holds.
38 00:03:29,983 –> 00:03:35,728 So if we have a sequence a_n, where we already know it’s convergent with limit “a”,
39 00:03:35,928 –> 00:03:40,132 then any subsequence we can choose is also convergent.
40 00:03:40,586 –> 00:03:48,063 Moreover we also know when we calculate the limit, which means we send k to infinity, we get the same result as before.
41 00:03:48,429 –> 00:03:53,986 Ok, maybe that’s not so surprising, but at this point you could ask: “Why do we even need subsequences?”
42 00:03:54,686 –> 00:03:58,643 and the answer is, they help us to analyse divergent sequences.
43 00:03:59,414 –> 00:04:04,061 For this let’s look at an example we already discussed at the beginning of this course.
44 00:04:04,614 –> 00:04:07,174 Namely (-1) to the power n.
45 00:04:07,571 –> 00:04:14,653 On the number line this means that we hit the point 1 infinitely many times and also the point -1.
46 00:04:15,486 –> 00:04:19,078 So clearly this is a sequence that is not convergent.
47 00:04:19,714 –> 00:04:23,241 However we still find convergent subsequences.
48 00:04:23,614 –> 00:04:30,805 For example we could restrict ourself to the even indices. Which means we stay at 1 for the whole sequence.
49 00:04:31,200 –> 00:04:33,824 Clearly that’s a convergent sequence
50 00:04:34,643 –> 00:04:37,374 and we immediately see the limit, which is 1.
51 00:04:38,014 –> 00:04:41,195 Of course we can also consider the odd indices.
52 00:04:41,395 –> 00:04:43,787 Which means we stay at -1 the whole time.
53 00:04:44,686 –> 00:04:49,313 This means that we get another subsequence, which is also clearly convergent.
54 00:04:50,029 –> 00:04:53,775 However with another limit, which is -1 in this case.
55 00:04:54,171 –> 00:05:01,107 Ok and now these limits we get for subsequences are called accumulation values for the sequence a_n.
56 00:05:01,500 –> 00:05:04,468 Indeed that will be our next definition.
57 00:05:05,243 –> 00:05:12,230 So any real number “a” is called an accumulation value of the corresponding sequence a_n
58 00:05:12,871 –> 00:05:17,512 if there is a subsequence a_n_k that has “a” as its limit.
59 00:05:17,712 –> 00:05:24,570 Ok, recalling the fact from above, you see that accumulation value is a generalization of the term limit,
60 00:05:25,043 –> 00:05:28,957 because a convergent sequence can only have one accumulation value.
61 00:05:29,062 –> 00:05:30,608 Namely the limit.
62 00:05:30,808 –> 00:05:35,236 However for a divergent sequence we could have different accumulation values.
63 00:05:35,436 –> 00:05:39,651 Of course this example here was very simple, so let’s look at another picture.
64 00:05:40,200 –> 00:05:44,450 So you could imagine a sequence that jumps to different parts on the number line,
65 00:05:44,650 –> 00:05:49,516 but then it goes back and it gets closer and closer to 4 different points.
66 00:05:50,257 –> 00:05:57,836 You don’t have a limit, because the sequence still jumps around, but you get closer and closer to different accumulation values
67 00:05:58,600 –> 00:06:05,320 or in other words, you could restrict yourself to sequence members that only live here and then you get a convergent sequence
68 00:06:06,271 –> 00:06:08,743 and of course you can do the same for the other 3 parts.
69 00:06:09,414 –> 00:06:16,137 Therefore you could say an accumulation value is just a point on the number line, where the sequence accumulates.
70 00:06:16,614 –> 00:06:21,693 Another thing i should really tell you is that there are a lot of different names for the same thing.
71 00:06:21,893 –> 00:06:26,061 For example, not so surprising, some people call it a “cluster point”
72 00:06:26,500 –> 00:06:29,713 or also “accumulation point” instead of “value”.
73 00:06:30,300 –> 00:06:34,046 However a little bit more confusing is the term “limit point”.
74 00:06:34,471 –> 00:06:40,461 Of course it makes sense, but please be careful. You could have many limit points for one sequence a_n.
75 00:06:40,661 –> 00:06:45,194 Therefore to avoid this confusion some people use the term “partial limit”.
76 00:06:45,671 –> 00:06:48,671 However I will stay at the term “accumulation value”.
77 00:06:49,429 –> 00:06:54,637 Ok, for the end of this video I give you an alternative definition of accumulation value
78 00:06:54,837 –> 00:07:04,357 or to put it in other words, “a” is an accumulation value of the sequence a_n, if and only if for all epsilon greater than 0,
79 00:07:04,471 –> 00:07:12,329 we have that the epsilon-neighbourhood of “a” contains infinitely many sequence members of our given sequence a_n.
80 00:07:12,929 –> 00:07:18,075 Of course this description fits perfectly with our name accumulation or cluster.
81 00:07:18,275 –> 00:07:23,868 Now, this statement is not so hard to show, you only have to recall that the epsilon-neighbourhood of “a”
82 00:07:24,457 –> 00:07:29,846 is given by the interval (a-epsilon, a+epsilon).
83 00:07:30,329 –> 00:07:38,066 Ok, then in the next video we will talk more about accumulation values and also talk about the Bolzano-Weierstrass theorem.
84 00:07:38,686 –> 00:07:43,193 This theorem will tell us something about the existence of accumulation values.
85 00:07:43,757 –> 00:07:46,735 Therefore I hope I see you there and have a nice day.
86 00:07:46,971 –> 00:07:47,843 Bye!
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Quiz Content
Q1: Consider the sequence $(a_n)_{n\in \mathbb{N}}$ given by $a_n = n^2$. Is there a convergent subsequence?
A1: No!
A2: Yes, only one.
A3: Yes, there are many.
Q2: Consider the sequence $(a_n)_{n\in \mathbb{N}}$ given by $a_n = \frac{1}{n}$. How many accumulation values does the sequence have?
A1: 0
A2: 1
A3: 2
A4: 3
Q3: Consider the sequence $(a_n)_{n\in \mathbb{N}}$ given by $$a_n = \begin{cases} 0 \text{ if } n \text{ even}\ 1 \text{ if } n \text{ odd and divisible by } 3\ 2 \text{ if } n \text{ odd and not divisible by } 3\\end{cases} $$ How many accumulation values does the sequence have?
A1: 0
A2: 1
A3: 2
A4: 3
Q4: Can a sequence have infinitely many accumulation values?
A1: Yes!
A2: No!
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Last update: 2025-01
