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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 %2,3y $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 √7a 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 F x %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, z2 % 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 }
