1. sect 3.1 #16 2. sect 3.2 #22 3. sect 3.3 #4 4. sect 3.4 #7 5. sect 3,5 #2 Recall that a "metric space" is a set X on which a distance function d is defined, which satisfies three axioms (positive definiteness, symmetry, and the triangle inequality). 6. Show that the following is a metric space: take X to be the open interval (0,1) and define the distance between two points of X to be d(x,y)=|x-y|/xy . (You must show that the three axioms are satisfied.) 7. We may make the plane R^2 into metric space by declaring that the distance between points (a,b) and (c,d) is |a-c|+|b-d|. (You do NOT have to prove that this one satisfies the axioms.) Describe the ball of radius 1 around the point (0,0). Show that set of points (a,b) having a^2+b^2 < 1 is an open set (a union of balls). If x_1, x_2, ... is a sequence of elements in a metric space X, we say this sequence converges to an element x of X if, for every r > 0, the ball B_r(x) contains almost all of the x_i . ("Almost all" means all but a finite number of exceptions.) 8. In the metric space of problem 6, is the sequence 1, 1/2, 1/3, 1/4 ... a Cauchy sequence? 9. Prove that limits of sequences in a metric space are unique. (That is, if the sequence converges to x, and it also converges to y , then x=y. Hint: if x and y are different, then the distance D=d(x,y) would be positive. Think about balls of radius D/2 ,)