By Steven G. Krantz

Do not get me unsuitable - Krantz is nice yet this is often primarily child Rudin - with out the proofs - that is type of like a bar with no beer.

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K +21)· k . N - (k 2 + . Sk-I,N 1 1) . N . We have succeeded in expressing Sk,N in terms of SI,N. N. Thus we may inductively obtain formulas for Sk,N for any k. ,N = 1 + 2 + ... N 2 2 2 = 1 + 2 +... N = 1 +2 + ... 3N __I.. ) 4 S4N=1 +2 +· .. 5 Operations on Series Some operations on series, such as addition, subtraction, and scalar multiplication, are straightforward. Others. such as multiplication. entail subtleties. 8 Let 00 :L>j j=1 00 and :L)j j=l be convergent series ofreal or complex numbers; assume that the series sum to limits Ci and p respectively.

J=1 Notice that each of the sets Sj is closed and bounded. hence compact. 7 of the last section. C is therefore not empty. The set C is closed and bounded, hence compact. 8 The Canlor set C has zero length, in the sense thaI [0. I] \ C has length 1. Idea of the Calculation: In the construction of SI. we removed from the unit interval one interval of length 3- 1• In constructing S2, we further removed two intervals of length 3-2 . In constructing Sj. we removed 2 j - 1 intervals of length 3- j Thus the total length of the intervals removed from the unit interval is 00 I)j-1 .

In JR. any open set U is the countable union of disjoint open intervals. 3. 3 It may be noied that the union of any number (finite or infinite) of open sets is open. The intersection of finitely many (but. in general. not of infinitely many) open sets is open. 1 Let U = (3,4) U (7,9). Then U is open. To illustrate this point. we take. for instance. 88 E U. 1 and see 0 that (x - E. 98) C S. 2 Closed Sets A set E c JR is called closed if its complement C E is open. Unlike an open set. which is simply a union of intervals.