NONSTANDARD ANALYSIS: HYPERRATIONALS AND
INTERMEDIATE VALUE THEOREM.
Roberto Ribeiro BALDINO[1]
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ABSTRACT: The ideas of infinite and
infinitesimal numbers are rigorously presented in the numerical field of
hyperrational numbers, without any appeal to propositional calculus.
Hyperrational numbers are built as equivalence classes of rational sequences in
a process parallel to the construction of real numbers. From the existence of
delta-stable ultra-filters we show that in the equivalence class of any
hyperrational number there is a Cauchy rational sequence; this leads to the
characterization of real numbers as equivalence classes of finite hyperrational
numbers. We present a proof of the intermediate value theorem for rational
functions that gives meaning to the spontaneous student's proof: " the
point where the function vanishes is the one that comes immediately after those
where it is positive".
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KEYWORDS: Nonstandard analysis;
hyperrational numbers; hyperreal numbers; construction of real numbers.
[1] Departamento de Matemática –Instituto de Geociências e Ciências
Exatas - UNESP – 13500-230 – Rio Claro – SP.