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1. Fill in the blanks to give a correct characterization of zero infimum for a non-empty set A of real numbers. a) AaEA, osa and inf A = 0 def b) 3aEAs.t. 2. Let (X,d) be a metric space and E a non-empty subset of X. Given any point PEX, consider the set of all distances from p to points of E: e Ap={d(p.e):eeE}. d(p.e) E Observe that Ap is a set of real numbers which is non-empty (since E # 0) and bounded below. It d(p,e)EA, e' therefore has an infimum, which we define to be the distance dE(p) from the point p to the set E: dE(p) = inf.eed(p,e) = inf{d(p,e) eeE} = inf Ap . i) Explain why condition (a) in question 1 above holds for the set A = Ap. ii) Give two equivalent characterizations, as indicated, of what it means for p to have distance zero from E. Restate condition (b) from question 1 in de(p)=0 terms of points of the set E. Hestate equivalently IN terms of balls about p. 0001 (Fair 2017) 3. With notation as in question 2. fill in the blanks to characterize each of the following types of point p in terms of the distance de(p) from the set E. i) p is a point in the closure of E dE(p) ii) p is an exterior point of E dE(p) 4. With notation as in question 2. consider any points x,ye X. i) Show that dE(x) d(x,y) + d(y,2) for all 2 E E. ii) Use (i) to give a lower bound for the set A. iii) Use (ii) and the definition of infimum to show that dE(x) - dE(y) < d(x,y) iv) Explain why you can conclude from (iii) that in fact (dg(x) - dg(y)| < d(x,y) for all x.yex. 5. Let (X,d) be a metric space, and a E X. Prove the following facts about 8-balls. Draw schematic diagram in each case to illustrate your argument. i) If pe B(a,8), then B(p,8) CB(a, 28). ii) If pe B(a,8), then there is a positive number T such that B(p,r) C B(a,8).

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