By Graeme L. Cohen

Designed for one-semester classes for senior undergraduates, this ebook ways themes at first via convergence of sequences in metric house. besides the fact that, the choice topological procedure can also be defined. purposes are incorporated from differential and indispensable equations, platforms of linear algebraic equations, approximation conception, numerical research and quantum mechanics.

Cover; Half-title; Series-title; name; Copyright; Contents; Preface; 1 Prelude to trendy research; 2 Metric areas; three The fastened element Theorem and its purposes; four Compactness; five Topological areas; 6 Normed Vector areas; 7 Mappings on Normed areas; eight internal Product areas; nine Hilbert area; Bibliography; chosen recommendations; Index.

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**Sample text**

And it is clear with this arrangement how the integers may be counted. It now follows that any other set is countable if it can be shown to be equivalent to Z. In fact, any countable set may be used in this way to prove that other sets are countable. The next theorem gives two important results which will cover most of our applications. The second uses a further extension of the notion of a union of sets, this time to a countable number of sets: if X±, X 2, . , are sets, then OO |^J Xk = {x : x G Xk for at least one k = 1, 2, 3, .

For which 0 is a cluster point. It follows that 0 is a cluster point for the sequence. The sequence { ( —l ) n} has a finite range: it has two constant subsequences, namely —1 , —1 , —1 , . . and 1 , 1 , 1 , . . , so —1 and 1 are cluster points for this sequence. The sequence 1 , ^ , 1 , ^ , 1 , | , . . has cluster points at 1 and 0 since 1 , 1 , 1 , . . is a constant subsequence and 0 is a cluster point for the range { 1 , . }. Obviously, a cluster point for a sequence need not be an element of its range.

9. 10, suppose £ > u and set 6 = |(£ — u). As £ is a cluster point for S , there must exist a point of S in (£ — <5, £ + <5). Let xo be such a point. Then xo > £ - <5 = £ - §£ + \u = §£ + \u > \u + \u = u; that is, xo > u. This contradicts the statement that x < u for all x E S, so it cannot be possible to have £ > u. Thus £ ^ u. It is similarly proved that I ^ £. 5 (1) Let S = {1 + ( 1 /^ ) — ( V n) : m) n E N }. Find inf S and sup£. (2) Suppose a nonempty point set S is bounded below. Show that inf S = —sup{—x : x E £ }.