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Title: Supremum and Infimum
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Series: Real Analysis
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Chapter: Sequences and Limits
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YouTube-Title: Real Analysis 6 | Supremum and Infimum
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Bright video: https://youtu.be/8Cyvdv7Sm2s
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Dark video: https://youtu.be/gfGynbfCgSw
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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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Print-PDF: Download printable PDF version
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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: ra06_sub_eng.srt
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Timestamps
00:00 Intro
00:14 Intervals
01:31 Maximum and upper bounds
02:19 Minimum and lower bounds
02:44 Definition of bounds
04:14 Examples
05:43 Definition Supremum and Infimum
08:40 Credits
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Subtitle in English
1 00:00:00,420 –> 00:00:03,410 Hello and welcome back to real analysis
2 00:00:03,980 –> 00:00:08,880 and as always first i want to thank all the people that support this channel on Steady or Paypal.
3 00:00:09,340 –> 00:00:13,170 In todays part 6 we will talk about the supremum and the infimum.
4 00:00:13,800 –> 00:00:18,600 However before we start with this lets look at some simple subsets of the number line.
5 00:00:19,220 –> 00:00:25,110 We can see the real numbers as this number line, because we know the real numbers are totally ordered.
6 00:00:25,730 –> 00:00:33,220 Therefore it makes sense to say we start with a number “a” and then we go to another number “b” which is greater or equal than “a”
7 00:00:34,120 –> 00:00:38,000 and now all the numbers in between i want to put in one set.
8 00:00:38,700 –> 00:00:46,400 Such a set is then called an interval where we use different brackets to denote if the endpoints are included or excluded.
9 00:00:46,830 –> 00:00:51,720 In this case here the set consists of all the points “x” that are inbetween “a” and “b”.
10 00:00:52,510 –> 00:00:55,900 However “a” itself is not in the set, but “b” is.
11 00:00:56,590 –> 00:00:59,590 Of course you can change that just by changing the brackets.
12 00:01:00,050 –> 00:01:01,430 So please remember that.
13 00:01:01,460 –> 00:01:04,950 We use the round brackets, the parentheses to denote exclusion.
14 00:01:05,600 –> 00:01:10,090 With this you see, intervals are just special subsets of the real number line.
15 00:01:10,470 –> 00:01:13,200 Please note we also use the interval notation
16 00:01:13,220 –> 00:01:15,220 when we don’t have a bound on one side.
17 00:01:15,830 –> 00:01:22,420 There we simple use the symbol infinity to denote that we start with “a”, but then we go as far as we want.
18 00:01:23,210 –> 00:01:27,090 So this interval consists of all the points “x” that are greater or equal than “a”.
19 00:01:28,010 –> 00:01:31,410 And in the same way the symbol minus infinity is used.
20 00:01:32,030 –> 00:01:38,720 Ok, now you should see these intervals are very nice subsets of the real numbers, but of course not the only ones.
21 00:01:39,350 –> 00:01:44,840 For example we could consider a set M that consists of all the points here on the number line.
22 00:01:45,140 –> 00:01:48,140 It’s much more complicated than just an interval,
23 00:01:48,150 –> 00:01:51,650 but still we can talk about nice properties of the subset M.
24 00:01:52,330 –> 00:01:56,250 For example this point here that is the farest on the right hand side of M
25 00:01:56,350 –> 00:01:58,100 We could call the maximum of M.
26 00:01:58,770 –> 00:02:01,100 In short we would just write “max M”.
27 00:02:02,030 –> 00:02:04,630 We could also call it an upper bound for the set M,
28 00:02:04,660 –> 00:02:08,650 because there is no other point in M that exceeds this number.
29 00:02:09,150 –> 00:02:13,150 However, this property holds for all the numbers here on the right hand side
30 00:02:14,150 –> 00:02:18,250 Hence every point in this region here is called an upper bound for M.
31 00:02:18,800 –> 00:02:22,900 and then you might already guess, we have lower bounds here on the left.
32 00:02:24,130 –> 00:02:28,420 Now, what we also see on the left is that we don’t find a minimal element in M.
33 00:02:29,130 –> 00:02:33,120 Simply because the boundary point here is not an element in M.
34 00:02:33,830 –> 00:02:35,830 Hence if we just live in the set M
35 00:02:35,840 –> 00:02:39,220 we can go to the left, but we never find a smallest one.
36 00:02:40,270 –> 00:02:43,270 Ok, so lets put all of this into a definition.
37 00:02:44,230 –> 00:02:49,220 So for any subset M we call a real number “b” an upper bound for M
38 00:02:49,740 –> 00:02:53,340 if all elements in M are less or equal than this “b”.
39 00:02:53,980 –> 00:02:56,170 There are two important things you should really note here.
40 00:02:56,230 –> 00:03:00,610 First “b” is just a real number. It does not have to lie in M.
41 00:03:01,220 –> 00:03:05,160 and second we could have a lot of different upper bounds for the set M.
42 00:03:05,930 –> 00:03:09,130 Therefore we just talk about an upper bound for the set M.
43 00:03:09,960 –> 00:03:14,060 Now in the same way a real number “a” is called a lower bound for M
44 00:03:14,590 –> 00:03:18,590 if for all “x” in M. “x” is greater or equal than this “a”.
45 00:03:19,260 –> 00:03:22,850 Now in the case that for a set M such an upper bound exist
46 00:03:22,920 –> 00:03:24,920 we call the set bounded from above.
47 00:03:25,780 –> 00:03:29,280 and in the case a lower bound exist we call it bounded from below.
48 00:03:30,120 –> 00:03:31,130 and you might already guess.
49 00:03:31,160 –> 00:03:34,250 If we have both things we just call the set bounded
50 00:03:35,030 –> 00:03:39,930 Ok, lets also fix the other notations we use namely maximum and minimum.
51 00:03:40,380 –> 00:03:45,250 Indeed this is very simple. If we have an upper bound “b” for the set M
52 00:03:45,270 –> 00:03:48,640 and this “b” is also an element of M.
53 00:03:48,650 –> 00:03:51,940 Then “b” is called a maximal element of M.
54 00:03:52,680 –> 00:03:57,180 and here we can show that we can only have at most one maximal element.
55 00:03:57,740 –> 00:04:00,840 Hence the notation we use “max M” is justified.
56 00:04:01,270 –> 00:04:04,070 Now the same definition we also have for a lower bound
57 00:04:04,080 –> 00:04:09,180 which is also in the set M and then we call it a minimal element.
58 00:04:09,960 –> 00:04:13,560 and the notation we use there is simply given by “min M”
59 00:04:14,040 –> 00:04:17,440 Ok, to get an idea of this lets look at a concrete example.
60 00:04:18,029 –> 00:04:23,020 Here the set M should be the interval that starts with 1 and ends with 3.
61 00:04:23,460 –> 00:04:26,040 and both numbers are included in the set.
62 00:04:26,760 –> 00:04:30,350 Therefore we immediately see the maximum of M is the number 3
63 00:04:30,590 –> 00:04:32,780 and the minimum of M is the number 1.
64 00:04:33,260 –> 00:04:37,050 However, when we now consider the set where we exclude both numbers
65 00:04:37,090 –> 00:04:41,080 we see that maximum and minimum don’t exist.
66 00:04:41,630 –> 00:04:46,520 In this case these two end points can’t be described by these simple descriptions.
67 00:04:47,270 –> 00:04:52,120 There we really need a new idea and this leads us to supremum and infimum.
68 00:04:52,680 –> 00:04:56,870 In order to define these two new things lets look at the number line again.
69 00:04:57,690 –> 00:05:02,780 So here you see our interval 1, 3. Where 1 and 3 are not included.
70 00:05:03,440 –> 00:05:07,530 Now we know for example 5 is an upper bound for this set.
71 00:05:08,090 –> 00:05:13,170 However we can also choose a smaller one. For example 4 is also an upper bound.
72 00:05:13,790 –> 00:05:18,890 Then the general idea is that we push this upper bound as far to the left as we can.
73 00:05:19,610 –> 00:05:24,710 Therefore in this case we will find the number 3 as the lowest possible upper bound
74 00:05:25,340 –> 00:05:28,620 and that is exactly what we will call the supremum of M
75 00:05:29,340 –> 00:05:36,820 actually then the result will be that this works for any set M. So any subset of R and not just for intervals.
76 00:05:37,520 –> 00:05:42,200 But for this we really need to write down the correct definition for the lowest upper bound.
77 00:05:42,650 –> 00:05:46,430 So lets consider again any subset M of the real numbers
78 00:05:46,470 –> 00:05:50,170 and then a real number “s” is called supremum of M
79 00:05:50,470 –> 00:05:52,970 if it fulfils two properties.
80 00:05:53,530 –> 00:05:57,420 The first one you already know. You want that “s” is an upper bound for M.
81 00:05:58,100 –> 00:06:00,800 So this is exactly the definition we had above.
82 00:06:01,430 –> 00:06:07,120 and now together with the second property we should describe that we get indeed the lowest upper bound.
83 00:06:07,770 –> 00:06:13,450 and there the idea should be if we subtract a small number epsilon from this number “s”
84 00:06:13,460 –> 00:06:17,450 we should get out a number that is not an upper bound anymore.
85 00:06:18,070 –> 00:06:24,660 In other words on the right of this number “s - epsilon” we will find at least one element of M.
86 00:06:25,340 –> 00:06:30,330 and if this works no matter how small we choose the epsilon we have the lowest upper bound described.
87 00:06:30,960 –> 00:06:33,990 Now in formulars we would write there is some “x tilde”
88 00:06:34,020 –> 00:06:37,440 such that “s-epsilon” is less than this x tilde
89 00:06:38,020 –> 00:06:43,390 So please note this means exactly that “s-epsilon” is not an upper bound for M.
90 00:06:44,210 –> 00:06:48,290 and of course this needs to work for any epsilon greater than 0.
91 00:06:49,020 –> 00:06:54,200 Ok, in the case such an “s”, the supremum of M exist we use the notation from above.
92 00:06:54,840 –> 00:06:58,620 This makes sense, because you can easily show if the supremum exist
93 00:06:58,660 –> 00:07:01,750 there can only be one number “s” with these properties.
94 00:07:02,150 –> 00:07:06,630 However we want even more. Even in the case that a supremum does not exist
95 00:07:06,670 –> 00:07:08,870 because there is no upper bound for the set M
96 00:07:08,890 –> 00:07:10,870 We want to use this symbol here.
97 00:07:11,390 –> 00:07:15,470 Then supremum of M should just stand for the symbol infinity
98 00:07:16,060 –> 00:07:19,660 Then we quarantee that this notation always has a meaning.
99 00:07:20,330 –> 00:07:26,120 Maybe the only other strange case would be when you want to calculate the supremum of the empty set.
100 00:07:27,010 –> 00:07:32,950 Because there all numbers are upper bounds for this set we use the symbol minus infinity there.
101 00:07:33,910 –> 00:07:38,100 Ok there you have it. This is the whole definition for the supremum of a set M.
102 00:07:38,610 –> 00:07:42,310 and similarly we can write down the definition for the infimum.
103 00:07:42,950 –> 00:07:45,290 Of course this looks almost the same
104 00:07:45,320 –> 00:07:48,909 because the only difference ist now that we are on the left hand side here
105 00:07:49,020 –> 00:07:50,640 and increase the lower bounds.
106 00:07:51,500 –> 00:07:54,900 Therefore this “l” should be the greatest lower bound for M.
107 00:07:55,660 –> 00:08:01,240 So what you should see here is the only difference to the supremum is the directions of the inequalities.
108 00:08:01,820 –> 00:08:08,510 Also we have the similar notation if the infimum exist and otherwise we use the symbols infinity as well.
109 00:08:09,070 –> 00:08:14,450 So i already told you, if we write “sup M” or “inf M”, it always has a meaning.
110 00:08:14,850 –> 00:08:20,050 In fact this is a result that follows from the completeness axiom of the real numbers.
111 00:08:20,550 –> 00:08:24,450 So it would not be true if you were working just with rational numbers.
112 00:08:24,740 –> 00:08:27,740 So i would say, please remember the whole definition here
113 00:08:27,820 –> 00:08:31,910 and then in the next video i’ll explain connection to the completeness axiom.
114 00:08:32,200 –> 00:08:34,900 and of course i also show you more examples.
115 00:08:35,460 –> 00:08:38,750 Therefore i hope i see you there and have a nice day. Bye!
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Quiz Content
Q1: Which of the following subsets of $\mathbb{R}$ is bounded from above?
A1: $ [0,\infty) $
A2: $ \mathbb{N} $
A3: $ \mathbb{Z} $
A4: $ (-\infty, 10]$
A5: $ (-\infty, \infty)$
A6: $ { n^3 ~|~ n \in \mathbb{N} }$
Q2: Which of the following subsets of $\mathbb{R}$ is not bounded from below?
A1: $ [0,\infty) $
A2: $ \mathbb{N} $
A3: $ \mathbb{Z} $
A4: $ { n^3 \mid n \in \mathbb{N} }$
A5: $ (-20, 40] $
Q3: Consider $M \subseteq \mathbb{R}$ given by: $$ [1,10] \cup {13} \cup \Big{ \frac{1}{n} \mid n \in \mathbb{N} \Big}$$ Which number is not an upper bound for $M$?
A1: $ 20 $
A2: $ 17 $
A3: $ 15 $
A4: $ 13 $
A5: $ 10 $
Q4: Consider $M \subseteq \mathbb{R}$ given by: $$ [1,10] \cup {13} \cup \Big{ \frac{1}{n} \mid n \in \mathbb{N} \Big}$$ Which number is not a lower bound for $M$?
A1: $ -20 $
A2: $ -10 $
A3: $ -5 $
A4: $ 0 $
A5: $ 1 $
Q5: Which statement is false?
A1: $ \max[0,1] = 1 $
A2: $ \min[0,1] = 0 $
A3: $ \sup[0,1) = 1 $
A4: $ \sup[0,1] = 1 $
A5: $ \max[0,1) = 1 $
A6: $ \inf[0,1) = 0 $
A7: $ \inf(0,1) = 0 $
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Last update: 2025-01
