By Sterling K. Berberian

**Uploader's Note:** Ripped from SpringerLink.

The ebook bargains an initiation into mathematical reasoning, and into the mathematician's state of mind and reflexes. particularly, the elemental operations of calculus--differentiation and integration of capabilities and the summation of endless series--are equipped, with logical continuity (i.e., "rigor"), ranging from the genuine quantity process. the 1st bankruptcy units down designated axioms for the genuine quantity procedure, from which all else is derived utilizing the logical instruments summarized in an Appendix. The dialogue of the "fundamental theorem of calculus," the point of interest of the booklet, in particular thorough. The concluding bankruptcy establishes an important beachhead within the thought of the Lebesgue critical via common capacity.

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A metric space is said to be compact if every sequence in the space has a convergent subsequence (cf. 4, Exercise 12). 2), where a < b, and let d(x, y) = Ix - yl be the usual metric on X. Prove: X is compact if and only if X = [a , b]. 10. } (ii) If (X, d) is a compact metric space, show that the subset {d(x , y) : x, y EX} of JR is bounded. {Hint: For every pair of sequences (x n ) and (Yn) in X , the sequence of real numbers d(xn, Yn) has a convergent subsequence. } 9. Prove that if (an) is a monotone sequence in JR that has a bounded subsequence, then (an) is convergent.

7. 1. Definition. A cut (or Dedekind cut) of the real field lR is a pair (A, B) of nonempty subsets of lR such that every real number belongs to either A or B and such that a < b for all a E A and b E B. In symbols, A =J. 0, B =J. 0, lR = Au B, a < b (Va E A, bE B) . 2. Examples. If , E lR and A={xElR: x:S,}, B={XElR: x>,} then (A, B) is a cut of lR (note that A has a largest element but B has no smallest); the pair (**) A={XElR: x<,}, B={XElR: x~,} also defines a cut of lR (for which B has a smallest element but A has no largest).

2/3. 7. Prove that the sequence 1 1 1 an = - - + - - + · · · + - n+l n+2 n+n is convergent, to a limit :S 1. {Hint: Show that the sequence is increasing and bounded above by 1. } 8. Prove that the sequence (an) defined recursively by al = 1 and an+! = an + 1/ an (n = 1,2,3, ... ) is not bounded. 5. Subsequences 43 9. Prove that the sequence is convergent. } 10. jnn. } -+ O. {Hint: Show that an 1 and 11. Let (an) be a convergent sequence in lR. , let (J : lP -+ lP be injective and define bn = au(n).