-
Title: Sequences and Limits
-
Series: Real Analysis
-
Chapter: Sequences and Limits
-
YouTube-Title: Real Analysis 2 | Sequences and Limits
-
Bright video: https://youtu.be/1SguKALJji8
-
Dark video: https://youtu.be/xHwlUvghzTA
-
Ad-free video: Watch Vimeo video
-
Forum: Ask a question in Mattermost
-
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: ra02_sub_eng.srt
-
Download bright video: Link on Vimeo
-
Download dark video: Link on Vimeo
-
Definitions in the video: sequence of real numbers
-
Timestamps
00:00 Introduction
00:15 Definition of a sequence
01:50 Examples
06:06 Definition of convergence and divergence
08:55 Example and Archimedean property
11:56 Credits
-
Subtitle in English
1 00:00:00,600 –> 00:00:03,500 Hello and welcome back to real analysis
2 00:00:04,100 –> 00:00:08,700 and as always I want to thank all the nice people, that support this channel on Steady or PayPal
3 00:00:09,890 –> 00:00:12,400 Now, today we actually start with our analysis course
4 00:00:12,500 –> 00:00:15,000 by talking about talking about sequences and limits.
5 00:00:16,000 –> 00:00:19,500 Therefore let’s immediately start defining what a sequence is.
6 00:00:20,290 –> 00:00:24,000 More concretely we will say that we have a sequence of real numbers,
7 00:00:24,500 –> 00:00:28,190 when we have a map from the natural numbers into “R”.
8 00:00:28,600 –> 00:00:32,090 and most of the time such a map gets the name “a”
9 00:00:32,600 –> 00:00:36,700 In the same way also a map “a” from “N0” into “R”
10 00:00:36,700 –> 00:00:37,900 is called a sequence.
11 00:00:38,900 –> 00:00:42,100 Please recall. Here in the natural numbers we don’t include 0,
12 00:00:42,300 –> 00:00:44,300 but here in “N0” we do.
13 00:00:45,000 –> 00:00:47,400 So you see, the choice here just depends
14 00:00:47,400 –> 00:00:50,400 if you want to start counting with 1 or with 0.
15 00:00:50,600 –> 00:00:54,600 Now, when we deal with sequences we seldom write down such a map,
16 00:00:54,600 –> 00:00:57,480 but rather an infinite list of numbers.
17 00:00:57,680 –> 00:01:02,350 In other words when we put 1 into the function here we get “a1”.
18 00:01:02,550 –> 00:01:05,099 and we put the 1 in the index.
19 00:01:05,590 –> 00:01:09,290 and then we get a2, a3 and so on.
20 00:01:09,490 –> 00:01:11,280 Therefore please remember.
21 00:01:11,289 –> 00:01:15,000 Formally a sequence is a map, but we will use shorter notations.
22 00:01:15,900 –> 00:01:20,200 For example what you often will see is just “(an)”.
23 00:01:20,600 –> 00:01:23,590 and then to remind you that it is an infinite list
24 00:01:23,590 –> 00:01:26,590 we put “n in N” in the index here.
25 00:01:26,790 –> 00:01:32,000 Of course if we want to start with 0 here we will put “N0” in the index here.
26 00:01:32,200 –> 00:01:38,150 Alternatively, we could also put the starting number here as n=1
27 00:01:38,200 –> 00:01:40,600 and then remind us again it’s an infinite list,
28 00:01:40,650 –> 00:01:42,600 so we put infinity at the top.
29 00:01:42,900 –> 00:01:46,080 Now if from the context the starting number is clear
30 00:01:46,090 –> 00:01:49,200 we can just omit everything and just use the parentheses.
31 00:01:49,700 –> 00:01:52,200 Ok. I think that is enough about the notations.
32 00:01:52,300 –> 00:01:54,590 Let’s immediately look at some examples.
33 00:01:55,000 –> 00:01:58,970 The best way to describe a sequence is just to give a rule
34 00:01:58,990 –> 00:02:01,200 for all the sequence members “an”
35 00:02:01,600 –> 00:02:07,000 For example we could say that “an” = -1 to the power n.
36 00:02:07,600 –> 00:02:11,400 and then we can use the parentheses to denote the whole sequence.
37 00:02:11,600 –> 00:02:14,400 So here you see, this is a very simple sequence,
38 00:02:14,430 –> 00:02:17,500 because you see the first number is just -1
39 00:02:17,890 –> 00:02:19,890 and then the next is 1
40 00:02:20,530 –> 00:02:25,100 then -1 again and 1 again and so on.
41 00:02:26,010 –> 00:02:29,000 Therefore if you want to visualize this on the number line
42 00:02:29,150 –> 00:02:31,150 we would start at -1
43 00:02:31,350 –> 00:02:34,640 and then in the next step we jump to 1.
44 00:02:35,470 –> 00:02:38,700 Afterwards for the next step we jump back to -1.
45 00:02:39,340 –> 00:02:43,000 and then we know we continue this whole procedure with no end.
46 00:02:43,650 –> 00:02:45,050 Here you can remember
47 00:02:45,060 –> 00:02:49,460 This is always a good way to visualize a sequence on the number line.
48 00:02:49,990 –> 00:02:52,790 So you can see the sequence has time steps
49 00:02:52,800 –> 00:02:56,200 where we hit at each timestep a number on the number line.
50 00:02:57,030 –> 00:03:00,880 However this is not the only way to visualize a sequence
51 00:03:00,900 –> 00:03:03,970 because we already know it’s simply a map.
52 00:03:04,170 –> 00:03:07,800 Therefore you could also just draw the graph of this map.
53 00:03:08,530 –> 00:03:12,080 In other words we have a set in this coordinate system,
54 00:03:12,090 –> 00:03:16,770 but please remember we have as the domain just the natural numbers,
55 00:03:16,900 –> 00:03:19,150 but the codomain is the real numbers.
56 00:03:19,160 –> 00:03:22,020 Therefore on the y-axis we find “R”.
57 00:03:22,990 –> 00:03:26,990 In conclusion we don’t get a line in this plane. We just get points.
58 00:03:27,040 –> 00:03:31,030 For example for 1 we find -1. So a point here.
59 00:03:31,630 –> 00:03:36,420 Now this is very important. We have -1 as the value of this map.
60 00:03:37,000 –> 00:03:42,329 In the same way we have 1 as a value when we put 2 into the map.
61 00:03:43,000 –> 00:03:47,500 and then we can continue with 3, 4, 5 and so on.
62 00:03:47,700 –> 00:03:52,380 Now. The jumping we had before we now see here when we go to the right.
63 00:03:53,010 –> 00:03:57,680 Indeed we are very interested in what happens with the values of this sequence
64 00:03:57,700 –> 00:04:00,380 when we just continue here on the line.
65 00:04:00,880 –> 00:04:05,770 Roughly speaking what happens to the sequence when n goes to infinity?
66 00:04:06,220 –> 00:04:09,200 For this example you see not so much will happen,
67 00:04:09,210 –> 00:04:12,200 because we still jump between -1 and 1.
68 00:04:13,030 –> 00:04:15,530 It doesn’t matter how large our n is.
69 00:04:15,560 –> 00:04:17,620 The jumping is always the same.
70 00:04:18,089 –> 00:04:21,300 For this reason lets look at another example.
71 00:04:22,000 –> 00:04:26,550 Here our sequence should be defined by the rule “1 over n”.
72 00:04:27,750 –> 00:04:30,390 Now you immediately see this is way more interesting,
73 00:04:30,400 –> 00:04:32,400 because we get out different numbers.
74 00:04:33,020 –> 00:04:38,000 The first number is just 1, but then we have 1/2 then 1/3
75 00:04:38,280 –> 00:04:40,870 and then a lot of different fractions,
76 00:04:40,880 –> 00:04:43,920 because the denominator gets larger and larger.
77 00:04:44,410 –> 00:04:48,000 Now also this sequences we can visualize as a graph.
78 00:04:48,460 –> 00:04:50,750 There we just start with the value 1.
79 00:04:50,770 –> 00:04:56,750 Then the value 1/2, 1/3, 1/4 and so on.
80 00:04:56,760 –> 00:05:01,000 Here in fact something happens when we get larger and larger,
81 00:05:01,220 –> 00:05:04,700 because you see, we get closer and closer to 0.
82 00:05:05,010 –> 00:05:09,900 and this is what we will define soon as the limit of the sequence.
83 00:05:10,100 –> 00:05:14,500 So here we recognize that this sequence has such a nice property,
84 00:05:15,020 –> 00:05:18,700 but the sequence from before doesn’t satisfy such a rule.
85 00:05:19,490 –> 00:05:23,230 However, before we define the limit as a property of a sequence
86 00:05:23,260 –> 00:05:25,260 lets look at another example.
87 00:05:25,650 –> 00:05:29,730 Here I want to have the numbers that are given by the powers of 2.
88 00:05:30,150 –> 00:05:35,000 In other words we have 2, 4, 8, 16 and so on.
89 00:05:36,030 –> 00:05:38,300 Of course this is a very nice sequence,
90 00:05:38,300 –> 00:05:42,130 but now we want to look what happens again when we increase n
91 00:05:42,150 –> 00:05:44,050 so make it larger and larger.
92 00:05:44,380 –> 00:05:49,170 Then you see, the values we get out also get larger and larger.
93 00:05:49,980 –> 00:05:53,240 and indeed, there is no upper bound for the members in the sequence
94 00:05:53,260 –> 00:05:56,350 so we could say, that this limit should be infinity.
95 00:05:56,920 –> 00:06:00,100 But of course we don’t know what this means exactly
96 00:06:00,110 –> 00:06:02,400 so we have to clarify this as well.
97 00:06:03,050 –> 00:06:06,350 In order to do this, lets jump to our next definition.
98 00:06:07,130 –> 00:06:11,700 Here we will define the notion of a convergent sequence of real numbers.
99 00:06:12,270 –> 00:06:17,750 We say that a sequence “an” is convergent to a given number “a”
100 00:06:17,780 –> 00:06:23,200 if the sequence members “an” lie arbitrary close to “a” eventually.
101 00:06:23,750 –> 00:06:25,990 Now before we give the formal definition
102 00:06:26,000 –> 00:06:29,450 let’s visualize this idea on the number line.
103 00:06:29,660 –> 00:06:31,660 So here we have the point “a”
104 00:06:31,670 –> 00:06:34,670 and in green we have the epsilon-neighbourhood of “a”.
105 00:06:35,170 –> 00:06:38,100 This means for a given positive number epsilon
106 00:06:38,100 –> 00:06:43,000 we can look at the number “a + epsilon” and “a - epsilon”.
107 00:06:43,230 –> 00:06:47,650 and the whole region in green we call the “epsilon neighbourhood of a”
108 00:06:48,390 –> 00:06:51,600 Please note here that “a” doesn’t denote the sequence anymore,
109 00:06:51,610 –> 00:06:53,000 but just another number.
110 00:06:53,000 –> 00:06:54,600 This is just a common notation.
111 00:06:55,560 –> 00:06:59,460 Now if we want to have “a” as the limit of the sequence in some sense
112 00:06:59,480 –> 00:07:04,050 we really need to get closer and closer to “a” with the sequence members.
113 00:07:04,140 –> 00:07:06,300 Or in other words eventually
114 00:07:06,300 –> 00:07:10,000 all the sequence members have to lie in this epsilon-neighbourhood of a.
115 00:07:10,950 –> 00:07:13,450 Only finitely many can lie outside.
116 00:07:14,270 –> 00:07:16,500 For example here we could have “a1”
117 00:07:16,550 –> 00:07:18,000 and there “a2”,
118 00:07:18,510 –> 00:07:22,640 but at some point we will find an index “N” such that
119 00:07:22,650 –> 00:07:27,100 all the sequence members afterwards lie inside the epsilon-neighbourhood.
120 00:07:27,970 –> 00:07:30,600 So what you should see is that this is really needed
121 00:07:30,600 –> 00:07:32,980 if we want to make sense out of the sentence
122 00:07:32,990 –> 00:07:35,990 “an” gets closer and closer to the point “a”.
123 00:07:36,880 –> 00:07:39,130 Therefore formally we now would say
124 00:07:39,140 –> 00:07:46,040 There exits a “N” such that for all “n >= N”
125 00:07:46,140 –> 00:07:51,500 we have that the distance “an” to “a” is less than epsilon.
126 00:07:52,300 –> 00:07:55,400 and this distance we can measure with the absolute value.
127 00:07:56,150 –> 00:07:58,900 Please note. This means exactly the same as saying
128 00:07:58,920 –> 00:08:01,620 “an” lies in the epsilon-neighbourhood of a.
129 00:08:02,080 –> 00:08:04,080 However here you should see,
130 00:08:04,080 –> 00:08:10,070 this only describes the convergence to the point “a” if this works for any epsilon.
131 00:08:10,690 –> 00:08:14,580 So no matter of small the epsilon is this always works.
132 00:08:15,000 –> 00:08:17,300 Of course if we choose a smaller epsilon
133 00:08:17,350 –> 00:08:19,850 we may have to choose a bigger “N” here.
134 00:08:20,310 –> 00:08:22,000 In the end this doesn’t matter,
135 00:08:22,030 –> 00:08:27,020 because we still have infinitely many sequence members inside the epsilon-neighbourhood
136 00:08:27,030 –> 00:08:29,220 and only finitely many outside.
137 00:08:29,980 –> 00:08:33,679 and with this you have the full definition of convergence.
138 00:08:34,320 –> 00:08:38,100 Now the opposite of this we simply call divergence.
139 00:08:38,820 –> 00:08:42,900 So in the case we don’t find such a limit point “a” with the property above.
140 00:08:42,919 –> 00:08:45,100 We call the sequence divergent.
141 00:08:45,650 –> 00:08:47,650 We have already seen 2 examples,
142 00:08:47,660 –> 00:08:50,800 where it’s very obvious that we can’t find such an “a”.
143 00:08:51,380 –> 00:08:55,000 But of course you really should write down a correct proof for this.
144 00:08:55,750 –> 00:08:59,990 However maybe it’s more interesting to first look at our positive example.
145 00:09:00,660 –> 00:09:05,500 Or in other words. The sequence 1/n is convergent to 0.
146 00:09:06,220 –> 00:09:09,050 So our “a” from above is now just 0.
147 00:09:09,580 –> 00:09:13,400 We have already talked about this. Intuitively this makes sense.
148 00:09:13,870 –> 00:09:17,870 But now we are able to write down the formal proof of this statement.
149 00:09:18,550 –> 00:09:23,000 First you should note, since we have to show this statement for all epsilon
150 00:09:23,020 –> 00:09:26,010 we have to choose an arbitrary epsilon at the beginning.
151 00:09:26,480 –> 00:09:28,480 Therefore the first sentence should read
152 00:09:28,510 –> 00:09:32,500 Let epsilon be a real number that is greater than 0.
153 00:09:32,990 –> 00:09:36,990 and we also already know what the last sentence of the proof should be.
154 00:09:37,790 –> 00:09:43,500 Namely that we have that the distance “an” to 0 in this case is less than epsilon.
155 00:09:43,970 –> 00:09:49,500 and this should hold for all indices “n >= N”.
156 00:09:50,460 –> 00:09:54,940 Hence you see, the only thing that is missing here is the definition of “N”
157 00:09:54,950 –> 00:09:57,880 and the calculation to reach this result.
158 00:09:58,300 –> 00:10:01,260 Of course here we can already fill in some details,
159 00:10:01,280 –> 00:10:03,570 because we know the sequence “an”.
160 00:10:04,210 –> 00:10:07,130 First subtracting 0 doesn’t change anything
161 00:10:07,150 –> 00:10:09,540 so we have the absolute value of “an”.
162 00:10:10,160 –> 00:10:13,150 Which is of course simply 1/n.
163 00:10:13,730 –> 00:10:15,630 Now at this point you should see,
164 00:10:15,650 –> 00:10:20,640 because we have this inequality we have the other inequality for the reciprocals
165 00:10:21,330 –> 00:10:26,330 Or simply “1/n” is less or equal than “1/N”.
166 00:10:27,520 –> 00:10:29,520 Ok. Now with this we have filled in the calculation
167 00:10:30,280 –> 00:10:35,150 and now the only thing missing is that 1/N is indeed less than epsilon.
168 00:10:36,030 –> 00:10:40,750 Of course we can define “N” as we want so lets choose it so large that
169 00:10:40,750 –> 00:10:43,400 that “N” times epsilon is greater than 1.
170 00:10:44,080 –> 00:10:48,080 Hence you only have to ask yourself “is this really possible”.
171 00:10:48,790 –> 00:10:50,480 and the answer is yes.
172 00:10:50,490 –> 00:10:54,090 This is exactly our Archimedean property from our axioms.
173 00:10:54,700 –> 00:10:58,940 It just tells us that no matter how small a number epsilon is
174 00:10:58,940 –> 00:11:02,040 we can always exceed any number we want.
175 00:11:02,420 –> 00:11:06,050 Just by adding the number finitely many times.
176 00:11:06,630 –> 00:11:10,000 Therefore we just find a suitable “N” here.
177 00:11:10,950 –> 00:11:14,880 Now having this we can finally read the proof from left to right.
178 00:11:14,900 –> 00:11:16,350 and everything makes sense.
179 00:11:16,830 –> 00:11:19,930 and also of course our statement is proven.
180 00:11:20,880 –> 00:11:26,300 Ok. Here you have seen what your thinking process should be when you want to solve such a problem.
181 00:11:26,370 –> 00:11:29,270 You start with the things you need to put in
182 00:11:29,290 –> 00:11:31,800 and the things you want to show in the end.
183 00:11:32,390 –> 00:11:34,850 and then you try to fill in all gaps
184 00:11:34,910 –> 00:11:38,550 such that in the end you can read it from left to right.
185 00:11:39,120 –> 00:11:42,550 This means that sometimes you need to shift the things a little bit around
186 00:11:42,570 –> 00:11:44,570 to get your result in the end.
187 00:11:45,000 –> 00:11:46,900 Ok. I think this is good enough for today.
188 00:11:46,900 –> 00:11:48,640 I hope i see you in the next video,
189 00:11:48,660 –> 00:11:52,000 when we talk about the properties of a convergent sequence.
190 00:11:52,390 –> 00:11:55,380 So have a nice day and see you then. Bye!
-
Quiz Content
Q1: For which of the following statements is it not possible to find an example as a map $f: \mathbb{N} \rightarrow \mathbb{R}$? Here we use the notation: $$ \mathrm{Ran}(f) := { f(x) \mid x \in \mathbb{N} }$$
A1: $ \mathrm{Ran}(f) = \mathbb{N}$.
A2: $ \mathrm{Ran}(f) = \mathbb{R}$.
A3: $ \mathrm{Ran}(f) = { -1,1 }$.
A4: $ \mathrm{Ran}(f) = {1, \frac{1}{2}, \frac{1}{3}, \ldots }$.
Q2: How can you write the sequence $( (-1)^n )_{n \in \mathbb{N}}$ as an infinite list?
A1: $(1,-1,1,-1, \ldots)$.
A2: $(1,-1,-1,-1, \ldots)$.
A3: $(-1,1,-1,1, \ldots)$.
A4: $(1,1,1,1, \ldots)$.
Q3: Let $(a_n){n \in \mathbb{N}}$ be a sequence of real numbers and $a \in \mathbb{R}$. What is the correct definition of ‘The sequence $(a_n){n \in \mathbb{N}}$ is convergent to $a$’.
A1: $\forall \varepsilon > 0 ~~ \exists N \in \mathbb{N} ~~ \forall n \leq N ~:~ |a_n - a| < \varepsilon$.
A2: $\forall \varepsilon > 0 ~~ \exists N \in \mathbb{N} ~~ \forall n \geq N ~:~ |a_n - a| < \varepsilon$.
A3: $\exists \varepsilon > 0 ~~ \exists N \in \mathbb{N} ~~ \forall n \leq N ~:~ |a_n - a| < \varepsilon$.
A4: $\exists \varepsilon > 0 ~~ \forall N \in \mathbb{N} ~~ \forall n \leq N ~:~ |a_n - a| < \varepsilon$.
Q4: Let us prove that $(a_n){n \in \mathbb{N}} = (\frac{1}{n}){n \in \mathbb{N}}$ is convergent to $0 \in \mathbb{R}$. For this let $\varepsilon > 0$ and let us choose $N \in \mathbb{N}$ such that
A1: $N \cdot \varepsilon < 1$. Then $|a_n - a| = \frac{1}{n} \leq \frac{1}{N} < \varepsilon$.
A2: $N \cdot \varepsilon = 1$. Then $|a_n - a| = \frac{1}{n} \leq \frac{1}{N} < \varepsilon$.
A3: $N \cdot \varepsilon > 1$. Then $|a_n - a| = \frac{1}{n} \leq \frac{1}{N} < \varepsilon$.
A4: $N \cdot \varepsilon = 1$. Then $|a_n - a| = \frac{1}{n} \geq \frac{1}{N} < \varepsilon$.
-
Last update: 2025-01
