By Klaus W. Roggenkamp
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7. Examples. = 1. (1) limn→∞ n−1 n = ◦ 1 − 1ı = 1. In more Take an infinitely large ı and note that ◦ ı−1 ı = 1. detail, Euler wrote: “Since ı is an infinitely large number; therefore, ı−1 i Indeed, it is clear that the greater number substitute for ı, the closer the magnitude will become to one; if ı is greater than any assignable number then the fraction of ı−1 ı ı−1 ı will become equal to one” [109, p. 116]. (2) limn→∞ 2nn = 0. , 0 ≤ N/2N ≤ 2/(N − 1) ≈ 0. Hence, N/2N ≈ 0. e) = 0. Given a natural n, note that n 0 8 eliminates the unpleasant reversal of quantifiers for all standard functions and points. It is curious as well as luminous to observe also that Courant referred to the ε-δ-definition of continuity as the Cauchy definition whereas it is the definition of microcontinuity that mimics the words “an infinitely small increment given to the variable produces an infinitely small increment of the function itself” (cf. 5). At the same time, the ε-δ-definition of continuity, applicable to all functions with no exception, is only implicitly reconstructible from microcontinuity at a point by standardization. In other words, dom(Φ) := {x ∈ X : Φ(x) = ∅}. By analogy, the image of a correspondence is the image of its graph. 9. , Rel (X) and Rel (Y ). We may arrange the composite of X and Y , denoted by the symbol Y ◦X, collecting all couples (x, z) such that (x, y) ∈ X and (y, z) ∈ Y for some y: (∀ u)(u ∈ Y ◦ X ↔ (∃ x)(∃ y)(∃ z)(x, y) ∈ X ∧ (y, z) ∈ Y ∧ u = (x, z)). The inverse of X, in symbols X −1 , is defined as (∀ u)(u ∈ X −1 ↔ (∃ x)(∃ y)(x, y) ∈ X ∧ u = (y, x)). , (∀ u)(u ∈ IX ↔ (∃ x)(x ∈ X ∧ u = (x, x))).