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Number systems of analysis I
Number systems of analysis II
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absolutely convergent addition and multiplication algebraic apply arithmetic belong calculus called cardinal number Cauchy sequence commutative compact contains continuous functions countably infinite countably infinite sets definition denote derivative differentiable elementary equation equivalence classes equivalence relation example Exercises exists finite closed interval follows function defined given any positive Hence hyperreal number If/is implies infinite hypernatural number infinitesimal integer internal function irrational numbers least upper bound Lebesgue integral Lemma limit linear mathematical monad monotone increasing sequence natural numbers non-empty subset non-zero open interval open set ordered field ordered pairs particular Peano axioms pointwise polynomial positive integer positive real number power series Proof Prove rational number rational sequence real number real number system real sequences result Riemann integral ring set neN set theory Similarly standard statement Suppose symbol totally ordered totally ordered field ultrafilter uncountable write