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

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**Sample text**

The interval [a, b) stands for all real values between a and b, including a but not b. Similarly for (a, b] and (a, b), where the square bracket means that the endpoint is included and the parenthesis means that the endpoint is not included. An interval of the form (a, ∞) stands for all real numbers greater than a and (−∞, a) for all real numbers less than a. Notice that +∞ and −∞ are not real numbers but indicate that an interval has no upper or no lower bound. Exercise 11 (Answer page 244) Show that if a, b are observable and x ∈ [a, b], then the observable neighbor of x exists and is in [a, b].

We conclude this section by stating the final version of the Closure Principle. Closure Principle, Existential Version Given an internal statement with parameters p, p1 , . . , pk : If p1 , . . , pk are observable and there exists some object p for which the statement is true, then there exists some observable object p for which the statement is true. 6 Analysis with Ultrasmall Numbers Sets and Induction The last principle we need deals with the way one usually defines sets and functions. If P(x) describes some property of x and A is a given set, then there exists a unique set X such that, for all x, x ∈ X if and only if x ∈ A and P(x); we denote it {x ∈ A : P(x)}.

Pk there exist natural numbers ultralarge relative to p1 , . . , pk . ] Let ε be ultrasmall relative to 1; then there exists an ultrasmall number, say δ, which is ultrasmall relative to ε (hence also ultrasmall relative to 1). ε is ultrasmall relative to 1 0 1 0 ε 1 δ ε δ is ultrasmall relative to 1, ε 0 Basic Concepts 27 Exercise 19 (Answer page 245) Let p be observable relative to q1 , . . , q . Show that x is ultrasmall relative to p, q1 , . . , q if and only x is ultrasmall relative to q1 , .