Tutorial Note 10x
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Tutorial Note 10x

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6 pages
English

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MATH202 Introduction to Analysis (2007 Fall and 2800 Spring) Tutorial Note #10 Midterm Review (Real Number) Recall the two main theorems Theorem 1: (Supremum Limit Theorem) c is upper bound of S csupS Ther exist sw S,such tha tlimw c Theorem 2: (Infimum Limit Theorem) c is lowe rbound of S cinf STher exist sw S,such tha tlimw c Example 1 Compute the supremum and infimum of the set Sx 2y:x$%1,1',y$%2,5' Solution: (Step 1) First, sin%ce1 )*)1 and %2)+)5, thus we have $ '%5%1 2$%2')* 2+)1 25 11 So the upper bound and lower bound are 11 asnpde c-t5iv erely. (Step 2) To show supS11 (This maximum is obtained whexn 1 and y5) We construct ourw x 2y , ,Pick x 1% and y 5% , (notex $0,1' and y $%2,5)' , , 0Hence w 1% 2.5% /11% S and lim 1 11 2 Therefore by supremum limit theorem, we sgeupt S11 To show infS% 5(This minimum is obtained whexn %1 and y%2) , , $ 'Pick x 3%1 and y3%2 , (notex 3 0,1 and y3$%2,5)' , , 0So w 3%1 2.%2 /%5 S and lim 1 3%5 2 By infimum limit theorem, we gientf S% 5 Example 2 Find the supreme and infimum of the set 5 0S3x %y:x 6%2,37,y$0,3' 5 0 0(Step 1) Note th0at 8x 89 and 0)y )3 27 Hence 5 0 0%273 $0'%27)3x %y )3$9'%$0' 27 So the upper bound and lower bound are 27% a2n7d respectively 5(Step 2) To shows upS27 (It is obtained whexn 9x3 and y0) ,Pick x 3 and y (then x 6%2,37 and y $0,3') ...

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MATH202 Introduction to Analysis (2007 Fall and 2008 Spring) Tutorial Note #10 Midterm Review (Real Number) Recall the two main theorems Theorem 1: (Supremum Limit Theorem) c is upper bound of S  c  supS There exists w  S,such thatlim w c    Theorem 2: (Infimum Limit Theorem) c is lower bound of S  c  infS There exists w  S,such thatlim w c    Example 1 Compute the supremum and infimum of the set S  x  2y:x  $%1,1',y  $%2,5' Solution: %1 ) * ) 1%2 ) + ) 5 (Step 1) First, sinceand ,thus we have %5  %1  2$%2' ) *  2+ ) 1  2$5'  11 So the upper bound and lower bound are 11 and -5 respectively. (Step 2) supS  11x  1y  5 To show(This maximum is obtained whenand ) w x 2y We construct our   x 1 %y 5 %x $0,1' y $%2,5' Pick and, (note and)  
   w 1 % 2 .5 %/  11 % Slim  11 Hence and    supS  11 Therefore by supremum limit theorem, we get infS  %5x  %1y  %2 To show(This minimum is obtained whenand )   x  %1 y  %2 x  $0,1'y  $%2,5' Pick and, (note and)  
 w  %1  2 .%2 /  %5  Slim   %5 So and  infS  %5 By infimum limit theorem, we get
Example 2 Find the supreme and infimum of the set   S  3x% y: x %2,3,y  $0,3'  0  x 90 ) y) 3 27 (Step 1) Note thatand Hence  %27  3$0' % 27 ) 3x% y) 3$9' % $0' 27 %27 So the upper bound and lower bound are 27 andrespectively supS  27x 9  x  3y  0 (Step 2) To show(It is obtained whenand ) x 3 y x %2,3y $0,3' Pick and (then and)    w 3x %y 3$3' %. / 27 % <lim  27 Then ; and   supS  27 Therefore by supreme limit theorem, infS  %27x 0  x  0y  3 To show(obtained whenand x  0y  3 %x %2,3 y $0,3' Pick and (then and) = =  w  3x% y 3$0'% .3 %/ < lim  $%3' %27 So  and infS  %27 Therefore by infimum limit theorem, Example 3 Find the supreme and infimum of the set 1 ? A  x :x  $1,2'\Ā,y  $2,4' Ç Ā y x y (Note: Henceneed to be irrational andneed to be rational) 1 ) * ) 22 ) + ) 4 (Step 1) Note thatand ,so 65 11 1513 ? ? ?  1 )* ) 2   64 4+ 28 F ?F So the upper bound and lower bound areand respectively G ?H F supS x  2y  2 (Step 2) To show(when and) G L I K %x 2 %y 2  PickJ (But not) and√KJ N?   F ? w 2 So .2 %/ ;lim  S   and;O √ G .N / P
F supA  By supreme limit theorem,. G ?F infS x  1y  4 To show(when and) ?H  x  1 $once againnot x  1 ' andy  4 % Pick √ ?   ?F ? Then w  .1 /   1 ;lim S   √O and H?; H .HR / P ?F infS  So by infimum limit theorem, ?H Example 4 Find the supreme and infimum of the set S  x % y:x  Ā Ç 0,√3U, y Ā Ç $2,π' Hence we require x and y are both rational numbers. Solution: 0 ) * ) √3%2 ) + ) W (Step 1) Note thatand %√π  0 % √π ) * % y ) √3 % √2 Then √3 % √2%√π Hence the upper bound and lower bound areand respectively. supS  √3 % √2x  √3y  2 (Step 2) To show(when and) J XLY √Z[ L I  Āx √3 % ] K  PickJJ (not) andJLY J _ X^ √[w x %y % `2  Slim w √3 % √2 Then _ and ^  supS  √3 % √2 By supreme limit theorem, infS  %√πx  0y  π To show(when and) J L LYà I  ]  Ā PickJ andJJJ LY _  ^ w  x % y % ` Slim w  %√π Then _ and  ^ infS  %√π By infimum limit theorem,
Difficult situation: A)Unknown Set Example 5 supA  √7 Let A be the subset of real numbers which, find the supreme of the set B  x 7y: x, y  A Solution: x, y AsupA  √7x  √7y  √7 (Step 1) Noteand ,so and x 7y  √7U 7√7U  14√714√7 Therefore .So the upper bound is supB  14√7 (Step 2) To show supA  √7a  A Since .By supreme limit theorem, there exist a sequence such lim  √7 that x y a x, y A' Pick   (so   w x 7y a 7a B lim 14√7 Then   and supB  14√7 By supremum limit theorem, we conclude. x y y 7  or (Remark: Some students may set x √7 % ,which is not right since B is unknown set and we do not know what B exactly contains!) Example 6 infC  Let C be the subset of rational number which, find the infimum of the set D  p% q: p  C, q  0,1\Ā (Here q is an irrational number) Solution: p  Ā  p 0  q  1 (Step 1) Note thatand  p %q  ./ %1  %% Hence ,so the lower bound is  GG   infD  %p q  1 (Step 2) To show(when and) G  infC c  C Since ,by infimum limit theorem, there exist, such that lim j 
p c q 1 %p C q 0,1\Ā Pick  and (so and) √
    w p %q c %$1 %'  Dlim  % Then   and . √ G infD  % By infimum limit theorem,. G Example 7 A AA infA 2 Let, and be the subsets of real numbers such that, infA 6 infA 3S  Ak Ak A  and. Find the infimum of the set  (Step 1) x  Sx  Ax  Ax  A For any,then or or, x  2 but we must have, so the lower bound of S is 2. (Step 2) infS  2x  2A To show(happen whenby elements in) a A lima 2 By infimum limit theorem, there exists  such that . w a A Ak Ak Aw A kA kA Pick , since  , then   ) lim  2 and . infS  2 By infimum limit theorem, we conclude Example 8 supE  1 Let D be a subset of real number such that, find the supremum of the set  F S  x% 3y  2z: x Ā Ç $%1,e', y $1,3'\Ā,z  D' (Note that x is rational and y is irrational) (Step 1) %1 ) * ) ô1 ) + ) 3z  1 Since ,and  F Fx %3y  2z) e% 3$1'  2$1' e% 1 So e %1 The upper bound is (Step 2) supS  e% 2x  ey  1z  1 To show(when ,and )  F w x %3y 2z We construct   as follows: _ ^  x x Ā Ç $%1, e' For, we pick_^ y y 1  $1,3'\Ā For,we pick√ z e  Elim ô 1 For, by supreme limit theorem, there exist which , pick z e  _^  FF w x %3y 2z . /% 3 .1 /  2e Slim So  _, and  ^ √   ô %1 supS e% 1 . By supreme limit theorem,
Try to finish the following exercises, if you have any questions about the exercises, please feel free to find me. Exercise 1 Find the supreme and infimum of the sets by using limit theorem. x % 2y: x $2,3'and y  2,5' a) H  %: n ,m b) Ā Ç $1,2'    $1,2' Ç Ā y % z: x $2,3'\Ā, , y, z2 % x c) } |x % y|: x, y  $0, √3' Ç Ā d)V e %y: x $%2,ln3' Ç Ā, y  2, ∞' e)F  4x 2y :x  $2,5'\Āand y  $%3,2' f)Exercise 2 supA  3infA  %2 Let A be the non-empty subset of the real number, whichand . Find the supreme of the following sets  %y ,x, y A a) VNF 3x % 2y: x  A, y  Ā Ç $%2,2' b)6x %2y  z: x, y  A and z $0,2'\Ā c)$  1, π√5  x  y: x  A, y  Q Ç d) Exercise 3 a)Find the supreme and infimum of the following sets A  1 : n B  3 % p:p  Ā Ç $2,3'C  x  √2y:x, y 1,2\Ā , , b)Find the supreme and infimum of the set S  A k B k C Exercise 4 Find the infimum of the set V 1 S  x  Ā:{ sin ~converges }  T  b  \Ā:{ ln 1 ~ diverges,b  0 } 