-
Title: Sequences, Limits and Closed Sets
-
Series: Functional Analysis
-
YouTube-Title: Functional Analysis 4 | Sequences, Limits and Closed Sets
-
Bright video: https://youtu.be/2BpD3RX5EIE
-
Dark video: https://youtu.be/UHWKPqzFVYI
-
Ad-free video: Watch Vimeo video
-
Quiz: Test your knowledge
-
Dark-PDF: Download PDF version of the dark video
-
Print-PDF: Download printable PDF version
-
Exercise Download PDF sheets
-
Thumbnail (bright): Download PNG
-
Thumbnail (dark): Download PNG
-
Subtitle on GitHub: fa04_sub_eng.srt
-
Timestamps
00:00 Introduction
00:35 Sequence
01:01 Convergence
02:57 Closedness with sequences
04:12 Proof
-
Subtitle in English
1 00:00:00,471 –> 00:00:03,486 Hello and welcome back to functional analysis
2 00:00:03,615 –> 00:00:08,829 and as always I want to thank all the nice people that support this channel on Steady or Paypal.
3 00:00:09,214 –> 00:00:13,356 Today’s part 4 is about sequences, limits and closed sets.
4 00:00:14,214 –> 00:00:19,863 So you see, we’re still building up the foundations we need for doing functional analysis.
5 00:00:20,063 –> 00:00:27,678 Now, for metric spaces it turns out that one can use sequences to describe the properties of a metric space.
6 00:00:28,343 –> 00:00:35,112 You might already know, a sequence is just an ordered sets of points inside the metric space X,
7 00:00:35,771 –> 00:00:40,257 because we give the points names, we usually see a notation like this
8 00:00:41,157 –> 00:00:44,871 or in short you just write x_n with n in N.
9 00:00:45,643 –> 00:00:52,657 Of course in the formal way you would say, you just have a map from the natural numbers into the metric space X,
10 00:00:53,486 –> 00:00:59,991 because we can measure distances in a metric space, we can also talk about convergent sequences.
11 00:01:01,600 –> 00:01:08,729 A sequence x_n in a metric space (X,d), which means all the x_n come from a set X,
12 00:01:08,929 –> 00:01:15,240 is called convergent if there is a limit point we call x tilde.
13 00:01:15,800 –> 00:01:23,503 So what we want is that the members of the sequence get closer and closer to this limit point x tilde.
14 00:01:24,329 –> 00:01:30,617 Of course we already know how to measure such closeness. We can just use an arbitrary epsilon-ball.
15 00:01:31,200 –> 00:01:33,386 It should be centered at x tilde,
16 00:01:33,387 –> 00:01:40,863 but then no matter how small we choose the epsilon, almost all the members of the sequence should be inside this ball.
17 00:01:41,100 –> 00:01:48,712 More exactly this means we find an index such that members with a bigger index lie inside the epsilon-ball.
18 00:01:49,514 –> 00:01:54,324 Formally this reads then: for all epsilon greater 0,
19 00:01:55,029 –> 00:01:57,501 there exists an index capital N
20 00:01:58,243 –> 00:02:02,020 such that all the other indices greater or equal this capital N
21 00:02:02,943 –> 00:02:10,659 fulfill that the distance between x_n and our limit point x tilde is less than epsilon.
22 00:02:11,471 –> 00:02:17,848 In this case we then write x_n tends to x tilde if n goes to infinity.
23 00:02:18,048 –> 00:02:27,112 Alternatively we also use the limit notation. So we write limit n to infinity of x_n is equal to x tilde.
24 00:02:27,312 –> 00:02:34,491 If you see such notations please remind yourself that these are always given with respect to a metric d
25 00:02:34,691 –> 00:02:37,092 and of course we can use such notations,
26 00:02:37,129 –> 00:02:43,940 because in a metric space there can only be at most one x tilde that fulfills all of these things here.
27 00:02:44,643 –> 00:02:47,929 You can easily show that using the triangle inequality.
28 00:02:48,571 –> 00:02:56,693 Later we will see a lot of examples of convergent sequences. Therefore I would say we start here proving another important fact.
29 00:02:57,414 –> 00:03:04,139 Here we look again at a subset of the metric space X and we can say that this one is closed,
30 00:03:04,143 –> 00:03:10,645 if and only if we can’t leave the set from the inside by just using sequences.
31 00:03:11,229 –> 00:03:17,635 More exactly this means that a limit such a sequence could have must lie in the set A.
32 00:03:18,414 –> 00:03:26,363 This is fitting for our visualization, because closeness means that this boundary we see here already belongs to the set A.
33 00:03:26,457 –> 00:03:30,987 Writing that down gives us then: for every convergent sequence a_n,
34 00:03:31,187 –> 00:03:34,941 where a_n is just an element in capital A.
35 00:03:35,357 –> 00:03:42,589 So it’s a sequence inside A and usually one uses the sloppy notation writing it down as a subset of A.
36 00:03:43,243 –> 00:03:48,752 The important part here is of course, we have a convergent sequence, but only in the sense of the definition.
37 00:03:48,952 –> 00:03:54,444 So it’s a convergent sequence in the space X. So it has a limit inside X.
38 00:03:54,644 –> 00:03:57,146 However for the proposition we need more.
39 00:03:57,346 –> 00:04:04,011 We need that the limit, that we know exists, is also an element of A.
40 00:04:04,814 –> 00:04:11,972 Ok, so this is important because now we have a characterization for closed sets just by using sequences.
41 00:04:12,457 –> 00:04:16,439 Ok, then let’s do the proof, where we have to show 2 directions.
42 00:04:16,786 –> 00:04:21,979 I want to start with the one from right to left, because we can just do the contraposition here.
43 00:04:22,843 –> 00:04:26,477 Hence we have to assume that A is not closed.
44 00:04:27,271 –> 00:04:31,443 Hence by definition the complement is not open.
45 00:04:31,643 –> 00:04:36,356 Now please recall open means that for each point inside the set,
46 00:04:36,400 –> 00:04:42,768 you find an epsilon-ball around this point, such that the whole ball is inside A^c in this case.
47 00:04:43,571 –> 00:04:49,210 Not open then means that there is at least one x tilde where this is not the case.
48 00:04:49,543 –> 00:04:58,234 So you find an x tilde here on the boundary such that you can use any epsilon-ball, but you always hit points in A.
49 00:04:58,714 –> 00:05:06,425 Hence we conclude that we can construct a sequence a_n, where each a_n comes from one of these sets here,
50 00:05:06,625 –> 00:05:10,188 where for example we set epsilon as 1 over n.
51 00:05:10,388 –> 00:05:15,562 If we do that then we know that a_n converges to x tilde,
52 00:05:15,643 –> 00:05:20,473 because we get closer and closer to x tilde if we increase the index n.
53 00:05:21,000 –> 00:05:22,729 Ok, so this is our result.
54 00:05:22,755 –> 00:05:28,429 We now have a sequence in A, which is convergent, but its limit is not in A.
55 00:05:28,629 –> 00:05:35,404 Ok, so this was our proof by contraposition. Which means this implication is now finished.
56 00:05:35,757 –> 00:05:40,229 So, let’s do the other direction then, where I also want to use a contraposition.
57 00:05:40,771 –> 00:05:46,258 Here we now assume that there is a convergent sequence in A where the limit is not in A
58 00:05:46,900 –> 00:05:51,284 and as before I want to call this limit just x tilde.
59 00:05:51,286 –> 00:05:53,259 So we have the same picture as here.
60 00:05:54,043 –> 00:05:56,268 Now you know by the definition of the limit
61 00:05:56,468 –> 00:06:02,871 you can use any epsilon-ball around x tilde, you will always hit points of the sequence.
62 00:06:02,923 –> 00:06:04,607 So inside A
63 00:06:05,400 –> 00:06:11,664 and this then means that A^c, the complement of A is not an open set.
64 00:06:12,114 –> 00:06:17,238 However then by definition the set A itself is not a closed set
65 00:06:18,671 –> 00:06:23,611 and as you can see this closes our proof here. The proposition is correct.
66 00:06:24,114 –> 00:06:30,472 Maybe we also close the video here. In the next part we will talk about complete metric spaces
67 00:06:31,100 –> 00:06:36,691 and there we will discuss some examples and you’ll see why it is helpful to deal with sequences.
68 00:06:37,714 –> 00:06:42,029 So, thanks for listening and hopefully I see you there. Bye!
-
Quiz Content
Q1: Let $(X,d)$ be a metric space and $a \in X$. What is the correct definition of ‘The sequence $(a_n)_{n \in \mathbb{N}}$ is convergent to $a$’.
A1: $\forall \varepsilon > 0 ~~ \exists N \in \mathbb{N} ~~ \forall n \leq N ~:~ d(a_n,a) < \varepsilon$.
A2: $\forall \varepsilon > 0 ~~ \exists N \in \mathbb{N} ~~ \forall n \geq N ~:~ d(a_n,a) < \varepsilon$.
A3: $\exists \varepsilon > 0 ~~ \exists N \in \mathbb{N} ~~ \forall n \leq N ~:~ d(a_n,a) < \varepsilon$.
A4: $\exists \varepsilon > 0 ~~ \forall N \in \mathbb{N} ~~ \forall n \leq N ~:~ d(a_n, a) < \varepsilon$.
Q2: Let $(X,d)$ be a metric space. What is a property of a closed set $A$
A1: Every sequence in $A$ is convergent.
A2: Every sequence in $A$ is convergent and the limit lies in $A$.
A3: For every convergent sequence in $A$, the limit lies in $A$.
A4: For every bounded sequence in $A$, there is a convergent subsequence.
Q3: Let $X = (0,4]$ and $d(x,y) = \frac{1}{2}|x-y|$. Which of the following sequences $(x_n)_{n \in \mathbb{N}}$ is not convergent?
A1: $x_n = \frac{1}{n}$
A2: $x_n = 4 - \frac{1}{n}$
A3: $x_n = \frac{1}{2}$
A4: $x_n = 4$
-
Last update: 2024-10
