Definition. Let S be a subset of Rd. We say a E Rd is an accumulation point of S, if S contains points arbitrarily close to a, more precisely: for all r > 0, there is an x E S, such that 0 < |x -a| < r. In symbols, Vr > 0, Axe S,0 < |/x- a| <r or equivalently Vr > 0, (SnB,(a)) \ {a} /0. The set of all accumulation points of S is denoted by S'. We say a E S is isolated in S, if there are no points in S close to a, more precisely: for some r > 0, no other point x E 5, satisfies |x - a < / In symbols, =r > 0, Vx ES,x = a or |x -all 2 r or equivalently Er > 0,SnB, (a) = [a}. Example 10.1. For each of the sets in Example 5.2 find (a) the set of accumulation points (b) all the isolated points Definition. We say a sequence is divergent or diverges if it is not convergent. <Hence, (a,) is divergent if the negation of Eqn. (8.1) is true.> Problem 10.2. Write the negation of Eqn. (8.1) Example 10.3. a,, := (-1)" is divergent. If fact, suppose a,, - a. Let & := . For any N, there is an n > N, such that |(-1)" - al 2 4. Explain Example 5.2. Find the boundary of each set and explain why it is the boundary. (a) The half-open unit interval [0, 1) in R. (b) The half-open unit square [0, 1)= = [0, 1) x [0, 1) in IR2. (c) The open unit ball B, (0) in R2? <As is customary, 0 E R2 is shorthand for (0,0) e R2, and similarly in Rd .> (d) The set (0, 1) U (1, 2) in R. (e) The set 7 of all integers in R. (f) The set { { [ ke N } in R. (g) The set Q in R2. (h) The irrationals R \ Q in R. Example 8.1. For each the sequences below, guess the limit a and find an N, such that n > N = [an - al < 10. They are sequences in R, so || . || = |.| is the absolute value. (a) an = " (b)an = 37 (c) an := (-1)"-. Hint: Solve the inequalities - 16 < an - a < To for n.