Then there are infinitely many i where f ( i ) = 1 n and also infinitely many i where f ( i ) < 1 n + 1 (this happens for all i ∈ Y ∩ X n + 2 ∈ U). Suppose the intersection Y ∩ ( X n ⧹ X n + 1 ) is infinite for some n. By hypothesis, there is a Y ∈ U such that f is Cauchy on Y. Define a sequence f by setting f ( i ) = 1 n for i ∈ X n ⧹ X n + 1. Let X 1 = N, X 2, X 3, … be an inclusion-decreasing sequence of sets in U. Suppose every bounded sequence becomes Cauchy when restricted to a suitable U-dominant set Y ⊆ N. More precisely, we propose to factor the classical homomorphism Rather, we propose to exploit a concept that is a household word for most of the mathematical audience to a greater extent than either Fréchet filters or ends of functions, namely Cauchy sequences. The present text belongs to neither of the categories summarized in Section 1. 2 Refining the equivalence relation on Cauchy sequences Peano’s 1910 article seems to have been overlooked by Freguglia who claims “to put Peano’s opinion about the unacceptability of the actual infinitesimal notion into evidence”. Peano contradicts his contention of 1892, following Cantor, that constant infinitesimals are impossible. Commenting on Peano’s 1910 construction, Fisher notes that here The construction results in Peano’s partially ordered non-Archimedean ring Pe of ( 1.1) with zero divisors that extend R. For instance, fine ( f ) + fine ( g ) can be defined as fine ( f + g ), and fine ( f ) ⋅ fine ( g ), as fine ( f g ). ![]() Moreover, Peano defines operations on the fini. Similarly, if g ( n ) = n − 1, then 0 < fine ( g ) < r for every positive r ∈ R, since eventually 0 < g ( n ) < r. For instance, if f ( n ) = n, then fine ( f ) > r for every r ∈ R, since eventually f ( n ) > r. ![]() Extends the set of real numbers (included as constant sequences), is partially ordered, and includes infinite and infinitesimal elements.
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