Analysis with ultrasmall numbers by Hrbacek K., Lessmann O., O'Donovan R.

By Hrbacek K., Lessmann O., O'Donovan R.

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Pk . 9 Show that (relative to a fixed context): (1) {x ∈ R : x is not ultralarge} is not a set. (2) {h ∈ R : h is ultrasmall} is not a set. (3) For any x ∈ R, {y ∈ R : y x} is not a set. 10 Use the Principle of Mathematical Induction to prove that, for all n ≥ 1, 12 + 22 + . . + n2 = n(n + 1)(2n + 1) . 11 √ Let x1 = 2, xn+1 = 2 + xn . Use the Principle of Mathematical Induction to prove that xn ≤ 2, for all n ≥ 1. 12 Let a > 0; prove by induction that (1 + a)n ≥ 1 + na, for all n ∈ N. 13 Prove by induction: If 0 < ε < 1, then 0 < εn+1 < εn holds for all n ∈ N.

Q , then holds relative to f, a, q1 , . . , q if and only if it holds relative to q1 , . . , q . It follows that the statement is true in some context where the parameters f and a are observable, if and only if it is true in every context where the parameters f and a are observable. The last example is of great importance, and applies generally. By our convention, if a theorem does not specify the context of the relative concepts used in it, then we understand this context to be that of its parameters.

1) 1 + √ 1 ε δ δ √ √ (3) H + 1 − H − 1 (2) H +K H ·K 2+ε 2 (5) − 5+δ 5 √ 1+ε−2 (6) √ 1+δ (4) Exercise 8 (Answer page 243) √ Prove that if h is ultrasmall, then 1 + h 1. Exercise 9 (Answer page 243) √ Prove that if N is an ultralarge positive integer, then N N 1. Exercise 10 (Answer page 243) For x, y ∈ R define: x ∼ y if x − y is not ultralarge. Prove Rules 3 and 4 with ∼ in place of . Give an example of x, y ∈ R such that x y but not x1 ∼ y1 . The following principle characterizes the completeness of the real number system in terms of observability.

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