Introduction to analysis homework questionssee attached for questionsplease type your answerProblem 2.5.8: Let A = {0,1,2} and B = {-8,3,6}. Describe the
following sets by writing all elements: A + B, AB, A + (-1)A.
(Note that A – B is a set theoretic notation defined in Definition 1.1.1
part (5). In particular, A + (-1)B # A- B, and A – A = 0.)
Example 2.5.1: Please make clear of the distinction between inf and
min, and between sup and max. min[-1,8) = -1 = inf[-1,8). On the
other hand, we have sup[-1,8) = 8 while max[-1,8) does not exist.
Problem 2.5.1: Suppose S is a nonempty subset of R that is bounded
below. Let us denote the set{-s:E S} by -S.
(1) Prove that -S is bounded above. (Hence, sup(-s) exists by the
completeness axiom.)
(2) Prove that:
(i)-sup(-S)is a lower bound of S, and
(ii) if a E R is a lower bound of S, then a s-sup(-s).
Therefore, we have shown that if S is a nonempty subset of R that is
bounded below, then infs exists, namely infs = -sup(-s).
1. Problem 2.5.1. (Note that the set – defined here is the same set as (-1) defined in
Problem 2.5.8. So, you may want to read Problem 2.5.8 before you do this problem.)
Part (1) 5 points.
Part (2)
5 points
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