
Title: Introduction

Series: Real Analysis

Chapter: Sequences and Limits

YouTubeTitle: Real Analysis 1  Introduction

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Quiz: Test your knowledge

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Subtitle on GitHub: ra01_sub_eng.srt

Timestamps
00:00 Introduction
00:27 Topic of real analysis
01:31 Requirements
02:05 Axioms of the real numbers
03:54 Credits

Subtitle in English
1 00:00:00,000 –> 00:00:04,000 Hello and welcome to real analysis
2 00:00:04,200 –> 00:00:09,200 and first I want to thank all the nice people that support this channel on Steady or Paypal.
3 00:00:09,399 –> 00:00:11,399 This is part 1 of a new course
4 00:00:11,599 –> 00:00:15,590 where we talk about the analysis we can do with the real numbers.
5 00:00:15,790 –> 00:00:17,790 Before we start with it I can tell you
6 00:00:17,990 –> 00:00:21,980 you will always find the PDF versions of these videos and the quizzes
7 00:00:22,000 –> 00:00:24,680 where you can test your understanding of these topics
8 00:00:24,700 –> 00:00:26,700 in the description below.
9 00:00:26,790 –> 00:00:30,890 Now, the topic of real analysis is also known as calculus.
10 00:00:31,090 –> 00:00:33,090 But there are also a lot of other names.
11 00:00:33,090 –> 00:00:37,290 For example just analysis or infinitesimal calculus.
12 00:00:37,490 –> 00:00:39,490 So you might recognize your lecture here,
13 00:00:39,490 –> 00:00:41,690 but of course the name is not important.
14 00:00:41,890 –> 00:00:43,890 The goal of the course is important.
15 00:00:44,090 –> 00:00:50,090 This should be in the end the understanding of differential and integral calculations.
16 00:00:50,290 –> 00:00:54,280 Therefore in the end you will be able to understand everything
17 00:00:54,280 –> 00:00:56,480 about the symbol “df/dx”
18 00:00:56,680 –> 00:01:00,670 and also about the integral symbol “f dx”.
19 00:01:00,870 –> 00:01:05,170 With this you might already guess some of the topics we will cover here.
20 00:01:05,370 –> 00:01:08,370 First we will look at sequences of real numbers
21 00:01:08,570 –> 00:01:10,570 and then define what limits are.
22 00:01:10,870 –> 00:01:15,360 Afterwards we will talk about functions that are continuous.
23 00:01:15,560 –> 00:01:19,560 Then, usually it gets easier when we talk about derivatives.
24 00:01:20,760 –> 00:01:24,760 In the end integrals will close our real analysis course.
25 00:01:24,960 –> 00:01:26,960 So you see, we have a lot to cover here.
26 00:01:27,160 –> 00:01:31,150 Therefore I should tell you what the requirements are for this course.
27 00:01:31,350 –> 00:01:34,000 Indeed, this is very simple to say.
28 00:01:34,000 –> 00:01:36,000 I can put it into two words.
29 00:01:36,000 –> 00:01:38,200 You just have to know the real numbers.
30 00:01:38,400 –> 00:01:40,200 You don’t need much else,
31 00:01:40,200 –> 00:01:43,400 if you know how to calculate in the real numbers “R”,
32 00:01:43,400 –> 00:01:46,000 you know everything to follow this course.
33 00:01:46,200 –> 00:01:48,400 In the case you don’t know the real numbers.
34 00:01:48,410 –> 00:01:52,400 That is not a problem, because I have a whole video course about them.
35 00:01:52,600 –> 00:01:56,600 In “Start Learning Mathematics” you find everything you need.
36 00:01:56,600 –> 00:02:00,000 and the important videos are the ones about the real numbers.
37 00:02:00,200 –> 00:02:04,200 Therefore it might be sufficient to look at “Start Learning Reals”.
38 00:02:04,880 –> 00:02:08,400 For us here, we will take the axioms of the real numbers
39 00:02:08,600 –> 00:02:10,389 as our foundation.
40 00:02:10,590 –> 00:02:12,590 Indeed they are not so complicated.
41 00:02:12,590 –> 00:02:14,790 We just have a set with two operations:
42 00:02:14,780 –> 00:02:16,990 addition and multiplication
43 00:02:17,190 –> 00:02:22,000 and also with an ordering such that we have all these properties here.
44 00:02:23,000 –> 00:02:25,480 Roughly they tell us that we have a field of numbers
45 00:02:25,570 –> 00:02:27,680 that are also nicely ordered.
46 00:02:27,880 –> 00:02:33,000 Which simply means that we can visualize the real numbers as the number line.
47 00:02:33,200 –> 00:02:36,480 and the last property here is the completeness axiom,
48 00:02:36,680 –> 00:02:38,680 which talks about sequences.
49 00:02:38,880 –> 00:02:42,170 But don’t worry. We will talk about sequences a lot soon.
50 00:02:42,370 –> 00:02:45,370 Therefore you will understand this axiom in the end.
51 00:02:46,060 –> 00:02:48,660 However, what we need immediately from the beginning
52 00:02:48,760 –> 00:02:52,000 is the so called “absolute value” of a real number.
53 00:02:52,200 –> 00:02:56,190 So you see, it always gives us a positive number or zero.
54 00:02:57,000 –> 00:03:02,000 In other words, it measures the distance from 0 to the point x.
55 00:03:03,000 –> 00:03:07,190 Soon we will see it is very important that we can measure distances
56 00:03:07,190 –> 00:03:08,760 to do real analysis.
57 00:03:09,000 –> 00:03:11,580 That is what the definition of a limit
58 00:03:11,580 –> 00:03:14,000 or the definition of the derivatives needs.
59 00:03:14,390 –> 00:03:17,190 Now, what you really should know is what to do
60 00:03:17,180 –> 00:03:19,480 when you have two numbers in the absolute value.
61 00:03:20,000 –> 00:03:23,580 So they could be combined by multiplication or by addition.
62 00:03:24,000 –> 00:03:26,770 For the multiplication nothing special happens.
63 00:03:26,860 –> 00:03:29,960 You can just split it up into the two absolute values.
64 00:03:30,000 –> 00:03:32,740 However, for the addition this is not true.
65 00:03:32,900 –> 00:03:35,050 There we just have an inequality.
66 00:03:35,250 –> 00:03:38,250 and this is called the “triangle inequality”.
67 00:03:38,450 –> 00:03:40,450 and we will use that a lot.
68 00:03:40,650 –> 00:03:43,640 Ok! I think that is good enough for a short introduction here.
69 00:03:43,839 –> 00:03:47,200 In the next video we will start with our real analysis course
70 00:03:47,190 –> 00:03:49,400 by considering sequences.
71 00:03:49,600 –> 00:03:52,490 Therefore I hope I see you there and have a nice day.
72 00:03:52,500 –> 00:03:53,380 Bye!

Quiz Content
Q1: Which of these implications for real numbers $x \in \mathbb{R}$ is not correct?
A1: $x + 2 = 0$ $
\Rightarrow$ $x = 2$.A2: $x \cdot 2 = 1$ $
\Rightarrow$ $ x = \frac{1}{2}$.A3: $x^2 = 1$ $
\Rightarrow$ $ x = 1$.A4: $x \cdot (1) = 0$ $
\Rightarrow$ $ x = 0$.A5: $x > 0$ $
\Rightarrow$ $ x < 0$.Q2: Which of these set relations is false?
A1: $1 \in \mathbb{R}$.
A2: $\sqrt{2} \in \mathbb{R}$.
A3: $\frac{5}{9} \in \mathbb{R}$.
A4: $0.\overline{9} \in \mathbb{R}$.
A5: $\pi \in \mathbb{R}$
A6: $5 \in \mathbb{R}$
A7: None of them.
Q3: Which of these statements for the absolute value $\cdot$ and real numbers $x,y \in \mathbb{R}$ is not correct?
A1: $x + y = x + y$.
A2: $x \cdot y = x \cdot y$.
A3: $x + y \leq x + y$.
A4: $x \cdot y \leq x \cdot y$.