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Title: Completeness Axiom
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Series: Start Learning Reals
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Parent Series: Start Learning Mathematics
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Chapter: Real Numbers
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YouTube-Title: Start Learning Reals 2 | Completeness Axiom
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Subtitle in English
1 00:00:00,070 –> 00:00:03,410 Hello and welcome back to Start Learning Reals.
2 00:00:04,250 –> 00:00:06,230 And as always i want to thank all
3 00:00:06,230 –> 00:00:08,290 the nice people that support this channel on
4 00:00:08,290 –> 00:00:09,270 Steady or Paypal.
5 00:00:09,750 –> 00:00:12,170 Now today in part 2 we will talk
6 00:00:12,170 –> 00:00:14,430 about the axioms of the real numbers.
7 00:00:15,309 –> 00:00:17,270 And afterwards in the next video we will
8 00:00:17,270 –> 00:00:19,250 finally do the explicit construction.
9 00:00:20,050 –> 00:00:22,010 In order to make this work we will
10 00:00:22,010 –> 00:00:24,110 universally use the absolute value.
11 00:00:24,110 –> 00:00:26,930 We discussed it in the last video and
12 00:00:26,930 –> 00:00:28,890 i also told you that we use it
13 00:00:28,890 –> 00:00:29,930 to measure distances.
14 00:00:30,910 –> 00:00:32,890 And the notation is just given by 2
15 00:00:32,890 –> 00:00:34,650 bars around a rational number.
16 00:00:35,370 –> 00:00:37,790 Now one important property you can easily prove
17 00:00:37,790 –> 00:00:40,190 is just that it is compatible with the
18 00:00:40,190 –> 00:00:40,590 multiplication.
19 00:00:41,690 –> 00:00:43,090 Or to put it in other words you
20 00:00:43,090 –> 00:00:45,470 can just pull out the multiplication sign.
21 00:00:46,350 –> 00:00:48,910 And for this property the mathematician just says
22 00:00:48,910 –> 00:00:51,170 the absolute value is multiplicative.
23 00:00:51,170 –> 00:00:54,070 Then the next question would be what happens
24 00:00:54,070 –> 00:00:55,210 with the other operation.
25 00:00:56,170 –> 00:00:57,970 Here we can also pull out the addition,
26 00:00:58,370 –> 00:01:00,590 but then in general we just get an
27 00:01:00,590 –> 00:01:00,970 inequality.
28 00:01:02,230 –> 00:01:04,650 And for this the mathematician would say the
29 00:01:04,650 –> 00:01:07,250 absolute value fulfils the triangle inequality.
30 00:01:08,210 –> 00:01:10,070 And the reason why it is called triangle
31 00:01:10,070 –> 00:01:12,170 inequality we will just see later.
32 00:01:12,950 –> 00:01:14,950 Of course the important part here is that
33 00:01:14,950 –> 00:01:17,210 we still have an estimate when we measure
34 00:01:17,210 –> 00:01:17,790 distances.
35 00:01:18,740 –> 00:01:22,110 Furthermore please also recall that we discussed Cauchy
36 00:01:22,110 –> 00:01:23,670 sequences in the last video.
37 00:01:24,430 –> 00:01:27,030 A sequence of numbers we call xn is
38 00:01:27,030 –> 00:01:28,430 called a Cauchy sequence.
39 00:01:29,270 –> 00:01:31,930 If for all epsilon greater than 0 there
40 00:01:31,930 –> 00:01:34,490 exists an index capital N such that for
41 00:01:34,490 –> 00:01:38,090 all other indices nm greater than n, we
42 00:01:38,090 –> 00:01:40,510 have that the distance between sequence members is
43 00:01:40,510 –> 00:01:41,590 less than epsilon.
44 00:01:42,430 –> 00:01:44,730 To put it in other words the sequence
45 00:01:44,730 –> 00:01:47,470 members lie arbitrarily close eventually.
46 00:01:48,390 –> 00:01:50,710 Now you might already know that this looks
47 00:01:50,710 –> 00:01:53,310 similar to the notion of a convergent sequence.
48 00:01:53,950 –> 00:01:55,950 In this case we have a fixed number
49 00:01:55,950 –> 00:01:58,930 to which the sequence members get arbitrarily close
50 00:01:58,930 –> 00:01:59,410 eventually.
51 00:02:00,350 –> 00:02:02,190 So let’s call this number a and then
52 00:02:02,190 –> 00:02:04,710 we know it exists in a such that
53 00:02:04,710 –> 00:02:07,870 For each epsilon greater than 0 there exists
54 00:02:07,870 –> 00:02:10,450 an index n such that for all indices
55 00:02:10,450 –> 00:02:12,790 n greater than n, we have that the
56 00:02:12,790 –> 00:02:15,650 distance xn to a is less than epsilon.
57 00:02:16,470 –> 00:02:18,390 Of course in this case the number a
58 00:02:18,390 –> 00:02:20,250 is called the limit of the sequence.
59 00:02:21,150 –> 00:02:22,890 Later on we will talk more about the
60 00:02:22,890 –> 00:02:24,250 properties of such a limit.
61 00:02:25,130 –> 00:02:27,390 But now to get a visualisation let’s look
62 00:02:27,390 –> 00:02:28,590 at the number line again.
63 00:02:29,270 –> 00:02:31,310 For example a could be here.
64 00:02:31,310 –> 00:02:34,250 Then for a given epsilon we would have
65 00:02:34,250 –> 00:02:38,290 a minus epsilon here and a plus epsilon
66 00:02:38,290 –> 00:02:39,330 on the right hand side.
67 00:02:40,110 –> 00:02:41,810 And now we have here a nice region
68 00:02:41,810 –> 00:02:44,610 around a where each number in this region
69 00:02:44,610 –> 00:02:47,190 has a distance from a less than epsilon.
70 00:02:48,150 –> 00:02:50,750 Therefore this region is called the epsilon neighbourhood
71 00:02:50,750 –> 00:02:51,470 of a.
72 00:02:52,470 –> 00:02:56,210 Now the convergence property here guarantees that eventually
73 00:02:56,210 –> 00:02:59,130 all the sequence members lie in this epsilon
74 00:02:59,130 –> 00:02:59,490 neighbourhood.
75 00:03:00,330 –> 00:03:02,610 For example the sequence could start here.
76 00:03:02,930 –> 00:03:04,650 Then we have some members here and here
77 00:03:04,650 –> 00:03:05,150 and here.
78 00:03:05,410 –> 00:03:07,730 But then we have xn here.
79 00:03:08,490 –> 00:03:10,650 So this is x with capital N.
80 00:03:11,570 –> 00:03:14,610 And then all points that come afterwards lie
81 00:03:14,610 –> 00:03:16,530 inside our interval here.
82 00:03:17,530 –> 00:03:20,330 Hence only finitely many points can lie outside.
83 00:03:21,330 –> 00:03:22,490 However that’s not all.
84 00:03:22,750 –> 00:03:25,090 This whole thing works no matter how small
85 00:03:25,090 –> 00:03:26,670 we choose the epsilon at the beginning.
86 00:03:26,670 –> 00:03:29,550 The only thing that changes is how big
87 00:03:29,550 –> 00:03:30,910 the capital N has to be.
88 00:03:31,670 –> 00:03:33,970 Ok, with this let’s look at an example.
89 00:03:34,930 –> 00:03:37,030 Indeed the first example one sees most of
90 00:03:37,030 –> 00:03:39,450 the time is the sequence 1 over N.
91 00:03:40,290 –> 00:03:42,330 Here you should see the sequence members get
92 00:03:42,330 –> 00:03:44,910 closer and closer to the number 0.
93 00:03:45,890 –> 00:03:47,750 Ok, i think that should be clear and
94 00:03:47,750 –> 00:03:49,110 if you want to see a proof in
95 00:03:49,110 –> 00:03:51,050 a formal way we will do this later.
96 00:03:51,730 –> 00:03:53,770 Because now i want to talk about an
97 00:03:53,770 –> 00:03:54,550 important fact.
98 00:03:55,370 –> 00:03:58,070 So what is the relation between Cauchy sequences
99 00:03:58,070 –> 00:03:59,570 and convergent sequences?
100 00:04:00,430 –> 00:04:02,990 So you see both definitions look almost the
101 00:04:02,990 –> 00:04:05,230 same and even in the picture they would
102 00:04:05,230 –> 00:04:06,290 look indeed the same.
103 00:04:07,030 –> 00:04:08,910 The only difference is that in the case
104 00:04:08,910 –> 00:04:11,190 of a Cauchy sequence we don’t have the
105 00:04:11,190 –> 00:04:12,310 point A here given.
106 00:04:13,190 –> 00:04:15,390 Therefore the formulation is easier in this case
107 00:04:15,390 –> 00:04:17,190 because we don’t need the point A.
108 00:04:18,290 –> 00:04:20,550 Therefore we have the implication from the right
109 00:04:20,550 –> 00:04:22,330 hand side to the left hand side.
110 00:04:22,330 –> 00:04:26,250 A convergent sequence is always a Cauchy sequence.
111 00:04:26,770 –> 00:04:28,690 But not the other way around.
112 00:04:28,970 –> 00:04:31,850 The implication from left to right is only
113 00:04:31,850 –> 00:04:32,850 correct in R.
114 00:04:33,670 –> 00:04:35,990 In other words the real numbers have a
115 00:04:35,990 –> 00:04:38,770 very nice property the rational numbers miss.
116 00:04:39,690 –> 00:04:42,270 However the other implication is correct in Q,
117 00:04:42,470 –> 00:04:43,530 so we can prove it now.
118 00:04:44,230 –> 00:04:45,870 The thing we can use here is of
119 00:04:45,870 –> 00:04:49,210 course the triangle inequality for the absolute value.
120 00:04:49,210 –> 00:04:51,830 So for a Cauchy sequence we have to
121 00:04:51,830 –> 00:04:53,210 consider this expression.
122 00:04:53,630 –> 00:04:56,230 However for the convergent sequence we need to
123 00:04:56,230 –> 00:04:57,490 consider this expression.
124 00:04:58,330 –> 00:05:00,230 Of course this is no problem for us.
125 00:05:00,410 –> 00:05:02,750 We can just add and subtract A here.
126 00:05:03,410 –> 00:05:06,170 And now comes in the triangle inequality.
127 00:05:07,170 –> 00:05:09,170 So this simply means here we can pull
128 00:05:09,170 –> 00:05:10,050 out the addition.
129 00:05:10,950 –> 00:05:12,870 And with this we have established the connection
130 00:05:12,870 –> 00:05:15,670 between the convergent sequence and the Cauchy sequence.
131 00:05:16,500 –> 00:05:18,330 So of course this is the whole idea,
132 00:05:18,650 –> 00:05:20,910 but now we can formulate the formal proof.
133 00:05:21,790 –> 00:05:24,310 So first let’s fix a sequence xm, which
134 00:05:24,310 –> 00:05:26,510 should be convergent and let’s fix the limit
135 00:05:26,510 –> 00:05:27,090 as A.
136 00:05:27,790 –> 00:05:29,290 Then we know we want to show the
137 00:05:29,290 –> 00:05:31,310 Cauchy property for all epsilon.
138 00:05:31,710 –> 00:05:33,390 So let’s take an arbitrary epsilon.
139 00:05:34,490 –> 00:05:36,730 However now i want to define another epsilon
140 00:05:36,730 –> 00:05:38,270 called epsilon prime.
141 00:05:39,190 –> 00:05:41,790 So 2 times epsilon prime should be our
142 00:05:41,790 –> 00:05:42,790 original epsilon.
143 00:05:43,170 –> 00:05:45,290 Therefore we define it as epsilon half.
144 00:05:45,870 –> 00:05:48,370 Ok, at this point you should already see
145 00:05:48,370 –> 00:05:51,370 that this definition is useful, because we already
146 00:05:51,370 –> 00:05:54,090 know that in this expression we will add
147 00:05:54,090 –> 00:05:55,290 two epsilons.
148 00:05:56,290 –> 00:05:58,850 Therefore often in such proofs a definition in
149 00:05:58,850 –> 00:06:01,550 this way only makes sense after you have
150 00:06:01,550 –> 00:06:02,730 done all the calculations.
151 00:06:03,670 –> 00:06:05,750 Ok, now we can use that the sequence
152 00:06:05,750 –> 00:06:08,410 xm is convergent, so we know there is
153 00:06:08,410 –> 00:06:09,330 a capital N.
154 00:06:09,330 –> 00:06:12,810 Such that for all indices greater than N
155 00:06:12,810 –> 00:06:15,770 we know the distance xn to A is
156 00:06:15,770 –> 00:06:17,170 less than epsilon prime.
157 00:06:17,890 –> 00:06:20,230 So here we use the epsilon prime, because
158 00:06:20,230 –> 00:06:22,430 then we can make the conclusion we want.
159 00:06:23,330 –> 00:06:26,030 Namely we now can consider two indices that
160 00:06:26,030 –> 00:06:27,810 are both greater than the capital N.
161 00:06:28,510 –> 00:06:30,610 And then we just calculate the distance between
162 00:06:30,610 –> 00:06:31,830 both sequence members.
163 00:06:32,770 –> 00:06:35,210 Obviously now we want to use our calculation
164 00:06:35,210 –> 00:06:35,910 from above.
165 00:06:36,810 –> 00:06:39,030 And now by assumption for the indices we
166 00:06:39,030 –> 00:06:41,830 know this is less than epsilon prime and
167 00:06:41,830 –> 00:06:43,690 this is less than epsilon prime.
168 00:06:44,550 –> 00:06:47,430 So both things together are less than 2
169 00:06:47,430 –> 00:06:48,630 times epsilon prime.
170 00:06:49,190 –> 00:06:52,150 Which is by our definition just the original
171 00:06:52,150 –> 00:06:52,750 epsilon.
172 00:06:54,110 –> 00:06:56,130 And with this we now can put everything
173 00:06:56,130 –> 00:06:56,590 together.
174 00:06:56,590 –> 00:06:59,970 So for any epsilon greater than 0 we
175 00:06:59,970 –> 00:07:01,070 find a capital N.
176 00:07:01,450 –> 00:07:04,450 Such that for all nm greater than N
177 00:07:04,450 –> 00:07:07,210 we find that the distance between both members
178 00:07:07,210 –> 00:07:08,670 is less than epsilon.
179 00:07:09,550 –> 00:07:12,270 And that’s exactly the definition of a Cauchy
180 00:07:12,270 –> 00:07:12,850 sequence.
181 00:07:13,830 –> 00:07:16,290 With this we have proven the implication we
182 00:07:16,290 –> 00:07:17,110 wanted to prove.
183 00:07:18,070 –> 00:07:20,710 However for the real numbers we also want
184 00:07:20,710 –> 00:07:21,830 the other implication.
185 00:07:21,830 –> 00:07:24,790 And one solution to get this is just
186 00:07:24,790 –> 00:07:26,970 to take it as an axiom for the
187 00:07:26,970 –> 00:07:27,470 real numbers.
188 00:07:28,250 –> 00:07:31,390 So let’s call this the axiomatic solution, because
189 00:07:31,390 –> 00:07:33,290 here we don’t care about the construction.
190 00:07:34,290 –> 00:07:36,430 Indeed that is often the way one starts
191 00:07:36,430 –> 00:07:38,630 with the real numbers, because you just have
192 00:07:38,630 –> 00:07:40,530 a given rule set with which you can
193 00:07:40,530 –> 00:07:41,230 solve your problems.
194 00:07:42,090 –> 00:07:43,770 Of course what we need here is a
195 00:07:43,770 –> 00:07:45,670 non-empty set we call R.
196 00:07:46,470 –> 00:07:50,050 Together with two operations addition and multiplication.
197 00:07:50,830 –> 00:07:53,430 And an ordering less or equal.
198 00:07:54,270 –> 00:07:57,090 Now these things together we call the real
199 00:07:57,090 –> 00:07:59,910 numbers if they fulfil all the rules we
200 00:07:59,910 –> 00:08:00,230 want.
201 00:08:01,010 –> 00:08:03,330 Essentially we have all the properties we had
202 00:08:03,330 –> 00:08:05,710 for the rational numbers plus one additional.
203 00:08:06,570 –> 00:08:08,470 Ok, so let’s list the ones we already
204 00:08:08,470 –> 00:08:10,350 know and i want to start with the
205 00:08:10,350 –> 00:08:12,150 rule i call A for addition.
206 00:08:12,150 –> 00:08:15,210 It tells us that the set R together
207 00:08:15,210 –> 00:08:18,010 with the operation addition and the neutral element
208 00:08:18,010 –> 00:08:20,030 0 is an abelian group.
209 00:08:20,690 –> 00:08:24,690 So we have associativity, a neutral element, inverses
210 00:08:24,690 –> 00:08:26,530 and also commutativity.
211 00:08:27,510 –> 00:08:29,250 Next we have the same for the multiplication,
212 00:08:29,950 –> 00:08:32,590 however now we exclude 0 for the set.
213 00:08:33,370 –> 00:08:35,990 In particular 1 which is the neutral element
214 00:08:35,990 –> 00:08:39,070 with respect to the multiplication is not equal
215 00:08:39,070 –> 00:08:39,710 to 0.
216 00:08:39,710 –> 00:08:42,730 So please keep in mind we just list
217 00:08:42,730 –> 00:08:44,910 a set of rules here and the neutral
218 00:08:44,910 –> 00:08:48,130 elements in these two assumptions here just get
219 00:08:48,130 –> 00:08:48,950 some special names.
220 00:08:49,710 –> 00:08:51,510 Of course the names are chosen in such
221 00:08:51,510 –> 00:08:53,350 a way that they fit in with our
222 00:08:53,350 –> 00:08:54,370 other number sets.
223 00:08:55,290 –> 00:08:57,430 Ok, then the third rule i call D
224 00:08:57,430 –> 00:08:58,830 for distributive law.
225 00:08:59,810 –> 00:09:01,650 This is the same as always, it just
226 00:09:01,650 –> 00:09:03,770 connects multiplication and the addition.
227 00:09:04,570 –> 00:09:06,810 So at this point you already know these
228 00:09:06,810 –> 00:09:09,010 three rules together we call a field.
229 00:09:09,010 –> 00:09:11,910 Ok, then next i want to put everything
230 00:09:11,910 –> 00:09:13,550 about the order into O.
231 00:09:14,570 –> 00:09:16,210 So we have all the properties of an
232 00:09:16,210 –> 00:09:16,650 order ring.
233 00:09:16,950 –> 00:09:19,930 It’s also a total order and also compatible
234 00:09:19,930 –> 00:09:22,470 with the operations plus and times.
235 00:09:23,270 –> 00:09:25,710 And in addition we also have the so
236 00:09:25,710 –> 00:09:27,290 called Archimedean property.
237 00:09:28,070 –> 00:09:30,770 So now because you are very observant you
238 00:09:30,770 –> 00:09:33,510 know all these rules are fulfilled by the
239 00:09:33,510 –> 00:09:34,570 rational numbers Q.
240 00:09:35,410 –> 00:09:37,190 So you can watch all my videos where
241 00:09:37,190 –> 00:09:38,390 i talk about these properties.
242 00:09:39,010 –> 00:09:41,090 But of course you have waited for this
243 00:09:41,090 –> 00:09:44,050 for the last axiom we call the completeness
244 00:09:44,050 –> 00:09:44,570 axiom.
245 00:09:45,270 –> 00:09:48,210 There we just state that every Cauchy sequence
246 00:09:48,210 –> 00:09:50,430 is also a convergent sequence.
247 00:09:51,430 –> 00:09:53,590 And of course the distance is measured with
248 00:09:53,590 –> 00:09:56,190 the absolute value which is defined as for
249 00:09:56,190 –> 00:09:57,110 the rational numbers.
250 00:09:57,870 –> 00:10:00,030 And with this you have all the axioms
251 00:10:00,030 –> 00:10:01,170 of the real numbers.
252 00:10:01,850 –> 00:10:04,550 Of course the visualisation is again given by
253 00:10:04,550 –> 00:10:05,330 the number line.
254 00:10:05,330 –> 00:10:09,010 However now it’s the complete, the whole, the
255 00:10:09,010 –> 00:10:11,590 full number line without any holes.
256 00:10:12,470 –> 00:10:14,330 Ok, then in the next video i show
257 00:10:14,330 –> 00:10:16,510 you how we can calculate with all these
258 00:10:16,510 –> 00:10:17,590 rules very nicely.
259 00:10:18,430 –> 00:10:20,290 And afterwards i show you how we can
260 00:10:20,290 –> 00:10:22,430 actually construct this number set.
261 00:10:23,230 –> 00:10:25,070 Therefore i hope i see you there and
262 00:10:25,070 –> 00:10:25,790 have a nice day.
263 00:10:26,170 –> 00:10:26,570 Bye.
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Quiz Content
Q1: This is how the quizzes for the topics look like. You should start with the next video :)
A1: Yes!
A2: No!
A3: Never!
A4: Why???
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Last update: 2024-10
