By James R. Kirkwood

ISBN-10: 0534944221

ISBN-13: 9780534944223

Presents creation to research of real-valued services of 1 variable. this article is for a student's first summary arithmetic path. Writing type is much less formal and fabric provided in a manner such that the coed can strengthen an instinct for the topic and obtain a few adventure in developing proofs. The slower speed of the topic and the eye given to examples are supposed to ease the student's transition from computational to theoretical arithmetic.

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**Additional info for An Introduction to Analysis, Second Edition **

**Example text**

3, omitting proofs (see Exs. 2). An A or an N indicates that the infimum or supremum in question is attained or is not attained, respectively, in the domain of definition. 2A: inf/ = a [A], s u p / = a [A] (a real). 2 D, E : / not bounded above and not bounded below. 2F: inf/ = 0 [A], s u p / = 1 [N]. 2G: inf/ = 0 [A], s u p / = 1 [A]. 3 A: inf/ = 1 [A], sup / = | [A]. 3B: inf an = 0 [N], \αΛ not bounded above. 1. / / the real function f is defined on Si, then (i) m ^ inf/ ^ sup / ^ M, if m ^ f(x) ^ M for all xeS>, (ii) m g inf/, if f(x) ^ m for all xeS, (iii) sup / ^ M, if f{x)

If (iii) or (iv) holds, f is said to be strictly monotonie on J>x. Monotonie functions on Jx and J> are defined similarly. i) is increasing for n > X, and similarly in the other cases. We note, in particular, t h a t if f(n) is constant for all n i t is both increasing a n d decreasing according t o t h e definition. 1, a n d t h e other results of this section, can also be applied t o sequences. For example, if an is defined for all positive integers n, we say t h a t t h e sequence {an} is strictly increasing if am > an for all m > n > 0, a n d t h a t it is strictly increasing for n > X, where X ^ 0, if am* > an for all m > n > X.

If n > max(X 3 , X 4 ), we obtain f(n) < A + 1 < /(n), which is contradiction. Similarly, we cannot have /(τι) ->► ^4 and f(n) -> — oo as n ->* oo. 1 J to obtain a contradiction, since / cannot be both bounded and unbounded on J (X0) ; this proof does not extend without modification, however, to functions on 0t(X§) (see § 8). 2 with K — I, there exist numbers X5 and X6 such that f(n) > 0 for all n > X5, and — f(n) > 0 for all n > X e . This yields a contradiction for every n > max(X 5 , X 6 ). The argument used in part (i) will perhaps become clearer to the reader if he marks off the points A1} Az, Ax ± ε, Α2 ± ε on a line and observes that the two intervals of length 2 ε which are centred at Ax and A2 cannot overlap if ε ^ \{A2 — Ax).

### An Introduction to Analysis, Second Edition by James R. Kirkwood

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