0 Real Analysis - MATH20111

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1 0 Real Analysis - MATH20111 Warmup questions Most of you have seen limits, series, continuous functions and differentiable functions in school and/or in calculus in some form You might wonder why we are talking about them again now Here are a couple of questions related to these topics that will be answered in our course The difficulty in answering them is mostly not a lack of factual knowledge, but merely a lack of rigorous understanding of the fundamental notions limit, series, continuous function and differentiable function Answers to the warmup questions will be given in due course, but you might enjoy trying to tackle them right now I definitely recommend trying question (8) (1) What is lim n n n? What is lim n ( n+1 n)? (2) A series is an infinite sum of numbers But what does that mean? For example: What is ? Is it actually meaningful to ask this question? One might ask what is ? How would you argue for the answer this sum is 1? (3) A snail sits on one end of a rubber strap of 10 meters length After sunrise, the snail starts crawling towards the other end of the strap By sunset the snail made 1 meter of the strap and stops During the night, the snail rests and the strap is stretched uniformly by 10 meters On the next day, the snail is crawling 1 meter again and the procedure repeats Can the snail ever reach the end of the strap? (4) A popular explanation of continuous function reads: A function is continuous if I can draw its graph by a pen without taking the pen of the paper This is not a bad intuition of continuity, but insufficient in many cases For example: (a) Is the function f(x) = x sin( 1 x ) continuous at 0 when we set f(0) = 0? Can you decide this by drawing the graph of the function? (b) Is the boundary of a snow flake (the graph of) a continuous function? (5) Is every function continuous somewhere? In terms of the popular explanation of continuity in the previous item this asks: given a function, can you always draw at least a piece of the graph of the function? (6) Whatisthederivativeof x 3? Ifyouhavenotlearnedaruleforthisderivative, apply the definition of derivative (7) Give an example of a differentiable function y = f(x) such that dy dx is strictly positive at the number 2, but still the function is not increasing near 2 (8) Finally, a more subtle question: What is a real number? A popular answer is something like An expression of the form NXXX, where N is an integer and XXX stands for a possibly infinite string of digits If we accept this, then what is the difference between the number 0999 and the number 1?

2 Definition and first examples of sequences 1 1 Sequences 11 Definition and first examples of sequences A sequence is a list of real numbers a 1,a 2,a 3,a 4, indexed by natural numbers Examples of sequences: (i) 2,4,6,8, Here a 1 = 2,a 2 = 4,a 3 = 6 and a 4 = 8 (ii) 1, 1 2, 1 3, 1 4, Here a 1 = 1,a 2 = 1 2,a 3 = 1 3 and a 4 = 1 4 (iii) 0, 3 2, 2 3, 5 4, 4 5, (iv) 1, 2, 3 3, 4 4, (v) sin(1),sin(2),sin(3),sin(4), The precise definition of a sequence is the following: 111 Definition A sequence is a function N R, where N = {1,2,3,4,} is the set of natural numbers We write (a n ) n N for a sequences which maps n to a n The real number a n is called the n th term of the sequence (a n ) n N For example the 4 th -term in the sequence (iii) above, is Sequences in this sense are sometimes called infinite sequences It is important to notice that infinite here does not mean that the value set {a n n N} of the sequence (a n ) n N is infinite For example, the sequence 1,1, 1,1, (more precisely: (( 1) n ) n N ) is an infinite sequence but its value set is { 1,1} Often, sequences are communicated by its general term For example 1, 1,1, 1, is denoted by (( 1) n+1 ) n N and its general term is ( 1) n+1 The formal definition applies to our examples above as follows: (i) The sequence 2,4,6,8, is an indication of the function N R which maps a natural number n to 2 n: n 2 n So here the precise notation of the sequence is (2 n) n N and 2 n is its general term The value set is the set of even natural numbers 1 We will use this symbol to indicate a subtle fact, or to signal a warning that here common mistakes occur

3 2 Sequences (ii) The sequence 1, 1 2, 1 3, 1 4, is an indication of the function N R which maps a natural number n to 1 n : n 1 n So here the precise notation of the sequence is ( 1 n ) n N and 1 n is its general term The value set is the set { 1 n n N} Note (again) that the terms a n do not need to be different for different n In particular, for every real number r we have the constant sequence with value r: r,r,r,r, So here a n = r for every n N We can display sequences, more precisely the graph of a sequence (a n ) n N, as follows: R a 1 = a 8 a 4 = a 5 a 3 a 6 = 0 N a 2 a 7 = Elements (n,a n ) of the graph of (a n ) n N are indicated by a The value set {a n n N} is the set of all elements on the vertical line indicated by a 112 Definition A subsequence of a sequence (a n ) n N is any sequence of the form (a nk ) k N, where n 1 < n 2 < n 3 < itself is a strictly increasing sequence of natural numbers It is important to notice that in this case n k k for all k (as follows from a straightforward induction on k)

4 Convergent sequences 3 This definition looks more complicated as it is A subsequence of a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 can be thought of what is left from the original sequence by deleting some of its terms For example a 1 a 3 a 5 a 6 a 9 might be the beginning of a subsequence of (a n ) n N The only requirement is that the remaining list is not finite (that is, we have not deleted all but finitely many terms from the original list) The fancy double index notation of the definition, in the example above would be n 1 = 1,n 2 = 3,n 3 = 5,n 4 = 6,n 5 = 9; that is, the list n 1,n 2,n 3, simply is an enumeration of the remaining indices after the deletion 113 Examples Clearly, every sequence is a subsequence of itself The following are all subsequences of the sequence (a n ) n N : (i) (a 2n ) n N, so here n k = 2k This looks like a typo, however, (a 2n ) n N is exactly the same sequence as (a 2k ) k N (ii) (a 2 n) n N, so here n k = 2 k (iii) Take m N 0, where N 0 is defined as N 0 := N {0} = {0,1,2,3,} Then the sequence (a n+m ) n N is a subsequence of (a n ) n N Here n k = m+k Another notation for (a n+m ) n N is (a n ) n>m The following are all subsequences of the sequence ( 1 n ) n N: 12 Convergent sequences ( 1 n! ) n N, ( 1 n ) 1 n N and ( n 3 n+1 ) n N We start with a reminder of elementary properties of the modulus of real numbers 121 Rules about the absolute value Recall that for a real number a, the absolute value, or the modulus of a is defined as { a if a 0 a = a if a < 0 We have the following rules for all a,b R: (i) a b a b and a b, by definition of a (ii) The absolute value is multiplicative, ie a b = a b This is clear (iii) The triangle inequality holds: a+b a + b, because a+b a+ b a + b as b b as a a

5 4 Sequences and (a+b) = a+( b) a+ b as b b Replacing b by b in the triangle inequality gives a b a + b (iv) The following derived triangle inequality holds: a b a b, because a + b as a a a = (a b) + b a b + b, so a b a b, and by symmetry ( a b ) = b a b a = a b Replacing b by b in the derived triangle inequality gives a b a+b (v) For all ε > 0 we have a ε b a+ε a b ε, because a ε b means a b ε and b a+ε means (a b) = b a ε We are interested in the behavior of the terns of a sequence (a n ) n N for very large n In particular we want to know when the terms cluster at some real number The next definition is the most important one in this course 122 Main definition Let (a n ) n N be a sequence and let r R be a real number We say that (a n ) n N converges to r if for every real number ε > 0 there is some natural N N such that for every n N we have a n r < ε If this is the case we write a n r (n ) and also say a n has limit r as n tends to infinity, or a n tends to r as n tends to infinity or r is the limit of (a n ) n N A sequence (a n ) n N is called convergent if there is some r R with a n r (n ) Otherwise, (a n ) n N is called divergent 123 Discussion of the main definition Let us have a closer look at this definition Let r R be a real number and let (a n ) n N be a sequence (a) We claim that (a n ) n N converges to r if and only if foreveryrealnumberε > 0, then th -terma n liesintheopeninterval (r ε,r+ε) for all but finitely many n N r ε r r +ε To see this, note first that the expression a n r < ε in 122 is equivalent to a n (r ε,r+ε) (cf 121(v)) Now, the expression I For all but finitely many n N, a n is in (r ε,r+ε) is equivalent to the expression II There is some N N such that for all n N, a n is in (r ε,r+ ε) Convince yourself that I and II are indeed equivalent The idea here is, that the finite list of terms a 1,,a N 1 contains the exceptional terms, ie those terms for which a n is not in (r ε,r+ε)

6 Convergent sequences 5 (b) The natural number N in 122 depends on ε This becomes clear from the picture above If we shrink ε then we possibly have to increase N so that the terms a n with n N are all contained in (r ε,r+ε) (c) Let us illustrate (a) and (b) with the sequence ( 1 n ) n N We claim that ( 1 n ) n N converges to 0 How to prove this? Let us first write down the definition 122 of convergence for this sequence: For each ε > 0 we must find a natural number N N such that for all n N we have 1 n 0 < ε Note that r = 0 here We unwind this statement by first trying special values of ε: ε = 1 2 Then we have to find a natural number N N such that for all n N, 1 n < 1 2 Now we get a condition that we can work with, namely the condition: 1 n < 1 2 We know that this is true when n 3 and so here we have N: we choose N = 3 ε = 1 17 Then we have to find a natural number N N such that for all n N, 1 n < 1 17 Again we get a condition that we can work with, namely the condition: 1 n < 1 17 We know that this is true when n 18 and so here we choose N = 18 We see that the choice of N depends on ε Here the complete proof that ( 1 n ) n N converges to 0: Let ε > 0 We now do not specify a particular value of ε, but we think of ε as a given and fixed real number We must find a natural number N N (depending on ε) such that for all n N we have 1 n 0 < ε As in the special cases ε = 1 2, ε = 1 17 we obtain a condition that has to be satisfied, namely for all n N we have 1 n 0 < ε We rephrase 1 n 0 < ε to obtain the correct choice for N: 1 n 0 < ε is equivalent to 1 n < ε, that is, to 1 ε < n Hence if we choose N as a natural number with N > 1 ε, then n N implies n N > 1 ε and thus 1 n 0 < ε Again, the dependency of N on ε can be seen by the choice in this proof: we choose N as a natural number with N > 1 ε After this example, it is a good idea to read again items (a), (b) above! A proof of the convergence of a sequence directly from the definition as in item (c) above is called a proof from first principles (d) In compact form, definition 122 says that a n tends to r as n tends to if and only if ε > 0 N N n N a n r < ε If we want to stress the dependency of N on ε we also write ε > 0 N(ε) N n N(ε) a n r < ε 124 Remarks on wrong and imprecise formulations of convergence Let (a n ) n N be a sequence and let r be a real number Look at the following two statements which are frequently used to describe convergence of (a n ) n N with limit r (a) the terms a n of the sequence get progressively closer to the number r (b) For every real number ε > 0, all but finitely many a n lie in the open interval (r ε,r+ε) Both statements do not describe convergence:

7 6 Sequences (a) The main problem here is that the clause is imprecise and therefore cannot be applied when we want to check convergence Look at the following examples: Let a n = n, so the sequence looks like 2, 3 2, 4 3, 5 4, The terms a n get closer and closer to 0, but the sequence does not converge to 0 Let a n = n ( 1)n, so the sequence looks like 1,2, 1 3,4, 1 5,6, 1 7,8, Then the terms a n also get closer and closer to 0 in the sense that we can find terms of arbitrary small distance to 0 However the sequence doses not converge to 0 (or in fact to any number) Now let a n = 1+( 1)n n, so the sequence looks like 0,1,0, 1 2,0, 1 3,0, 1 4, Then the limit is 0, but each odd term is closer to the limit than is any even term So if we would take (a) as a definition, then (a n ) n N would not converge (b) This is flawed in the sense that it is not made clear what all but finitely many a n means: If all but finitely many a n means all but finitely many values a n, then the statement does not describe convergence Counterexample: a n = ( 1) n which looks like 1,1, 1,1, Here the set of all a n is { 1,1}; hence all but finitely many of the a n lie in every(!) interval So if we d choose this definition, then every real number would be a limit of this sequence On the other hand, if all but finitely many a n lie in means for all but finitely many n, a n lies in, then this is the correct definition Convergent sequences in everyday life: (a) If we want to measure surface areas (eg an ellipse) we approximate the area by small rectangles, write down the sum of the areas of these rectangles and hope that the sequence of these numbers progressively approximate the area (b) Given natural numbers n,m, we have a good intuition about what 2 n m is However, what shall we think of 2 2? Or better: How is 2 2 defined? We will answer this in due course (c) The decimal expansion of numbers is communicated by spelling out a sequence, eg π is 3, Which sequence do we actually mean here? If we would be able to write down the decimal expansion of π, our description would look like this: a 1 = 3 a 2 = 3,1 a 3 = 3,14 a 4 = 3,141 a 5 = 3,1415 a 6 = 3,14159 However, where does this sequence come from? Instead, we can do the following: For each n > 0 the interval (π 1 n,π) contains a rational number a n The sequence (a n ) n N obtained in this way, converges to π In fact, this argument has nothing to do with the special choice of the real number π and is valid for all real numbers In this sense we gave a precise description of

8 Convergent sequences 7 a sequence converging to π which does not tell us what π is! We ll come back to the precise definition of π later 125 Uniqueness of limits Let (a n ) n N be a sequences that converges to r R and to s R Then r = s Proof Suppose r s Wlog we may assume that r < s We choose ε = s r 2, which is half the distance of r to s In a picture: r r +ε = s ε s Since (a n ) n N converges to r, there is some N 1 N such that r ε < a n < r+ε for all n N 1 Since (a n ) n N converges to s, there is some N 2 N such that s ε < a n < s+ε for all n N 2 Hence for n N 1 and n N 2 we have s ε < a n < r+ε But this is not possible, since s ε = r +ε By the uniqueness of limits we may now also write if (a n ) n N is convergent with limit r lim n a n = r 126 Finite modification rule Let (a n ) n N be a sequences that converges to r R If (b n ) n N is another sequence with a n = b n for all but finitely many n, then also (b n ) n N converges to r Proof Firstly, we observe that the condition says precisely the same as a n = b n for all but finitely many n there is some K N with a n = b n for all n K In order to show that (b n ) n N converges to r we look into the definition of convergence 122: We have to pick some real number ε > 0 and we have to find some N N (depending on ε) such that, ( ) for all n N : b n r < ε So let us pick a real number ε > 0 Since (a n ) n N converges to r we already know that there is some N N such that for all n N we have a n r < ε Now if in addition n is bigger than K, then we know b n = a n So we choose N = max{n,k} and obtain ( )