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Math and science::Analysis::Tao::05. The real numbers

Real numbers, the construction from Cauchy Sequences

A real number is defined to be a new type of object, written as \( LIM_{n \rightarrow \infty} a_n \). This object has a (1 to many) correspondence to a Cauchy sequence \( (a_n)_{n=1}^{\infty} \). The Cauchy sequence is used to define an equivalence relation between real numbers: two real numbers \( LIM_{n \rightarrow \infty} a_n \) and \( LIM_{n \rightarrow \infty} b_n \) are said to be equal iff the corresponding sequences \( (a_n)_{n=1}^{\infty} \) and \( (b_n)_{n=1}^{\infty} \) are [...]