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QR - Q R Here follows one route from the rationals Q to the...

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Q R Here follows one route from the rationals Q to the reals R . The basic idea is to fashion each real number from the rational sequences that ‘converge’ to it; without having the actual real number in hand, we recognize ‘convergence’ from the Cauchy condition. Let R denote the set of all Cauchy sequences r = ( r n ) n =0 in Q ; here, to say that r is Cauchy is to say that for each ( rational !) ε > 0 there exists N such that q, p > N ⇒ | r q - r p | < ε ; such sequences are certainly bounded. Define addition and multiplication of rational sequences term-by-term: with the obvious notation, a + b = ( a n + b n ) n =0 , ab = ( a n b n ) n =0 . It is easy to check that these are binary operations that make R into a com- mutative ring, with additive identity and multiplicative identity the obvious constant sequences 0 and 1 (having constant terms 0 and 1 respectively). Let Z ⊂ R denote the set of all rational sequences z = ( z n ) n =0 that converge to zero (call them ’null’). It is easy to check that Z is an ideal in R : that is, Z
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