is bounded above and below and has a minimum and a maximum 2 S is bounded above

Is bounded above and below and has a minimum and a

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is bounded above and below and has a minimum and a maximum, (2) S is bounded above and below and has a minimum but not a maximum, (3) S is bounded above and below and has a maximum but not a minimum, (4) S is bounded above and below and has neither a maximum nor a minimum, (5) S is bounded below but not above and has a minimum, (6) S is bounded below but not above and does not have a minimum, (7) S is bounded above but not below and has a maximum, (8) S is bounded above but not below and does not have a maximum, (9) S is not bounded above nor is bounded below. Now show that each of these 9 cases leads to one of the 9 types described in the third ‘bullet’ in Chapter 2. I will do two as examples, the other seven are similar. For example, in case (3), let a = inf( S ) and b = sup( S ) = max( S ), then S ( a, b ]. Let y ( a, b ]. By the ‘Principle’ on page 18, there exists an x S with a x < y b . So y S since x, b S . So every y ( a, b ] is in S . Therefore ( a, b ] S , so S = ( a, b ], the third kind of interval. In case (8), let b = sup( S ), then S ( −∞ , b ). Let y ( −∞ , b ). Since S is not bounded below, y is not a lower bound for S , so there exists x S with x < y . By the ‘Principle’ on page 18, there exists a z S with y < z b . So y S since x, z S . So every y ( −∞ , b ) is in S . Therefore ( −∞ , b ) S , so S = ( −∞ , b ), the eighth kind of interval. square Homework 2. Text 5th Ed Section 3.2 number 3a,b,c, Section 3.3 numbers 1a-c, 3degi (prove these), 4e,i (prove these), 5, 6, 7, 8 [4th (and 3rd) Ed numbers: 12.1a-c, 12.3d,e,g,i (prove these), 12.4e,i (prove these), 12.5, 12.6, 12.7, 12.8.]
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22 MATH 3333–INTERMEDIATE ANALYSIS–BLECHER NOTES (A) Show that every nonempty subset of R which is bounded below has a greatest lower bound or inf (this falls out of the proof of Question 7(b) above (or 12.7 (b)) with k = 1). Also do case (4) and (9) in the proof of Theorem 3.13, in both directions (( ) and ( )). 3.4. Open and closed sets. In calculus you met some important definitions, something like the following: A neighborhood of x is an open interval centered at x . For example, (1 , 3) is a neighborhood of 2. Every neighborhood of x is of the form N ( x, ǫ ) = ( x ǫ, x + ǫ ) = { y R : | x y | < ǫ } , for some ǫ > 0. [Picture drawn in class.] We occasionally say that ǫ is the radius of N ( x, ǫ ). A deleted neighborhood of x is a neighborhood of x with the midpoint x removed: N ( x, ǫ ) = ( x ǫ, x ) ( x, x + ǫ ) = N ( x, ǫ ) \{ x } = { y R : 0 < | x y | < ǫ } . [Picture drawn in class.] An interior point of a set S is a point x S such that S contains some neighborhood of x . That is, ǫ > 0 s.t. N ( x, ǫ ) S . Or equivalently, ǫ > 0 s.t. ( x ǫ, x + ǫ ) S . The interior of a nonempty set S is the set of all interior points of S . We write this set as S or Int( S ). This may be the empty set (if S has no interior points). For example if A = [1 , 3) then A = (1 , 3).
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