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 equivalent Cauchy sequences.The set of real numbers is denoted \( \mathbb{R} \).
\( LIM_{n \rightarrow \infty}a_n \) is referred to as a formal limit. It is not the standard limit, but an object taking the form of a limit, and existing by definition.
The next step (for another card) is to ensure via proof that the equivalence relation that has just been defined obeys the axioms of equality (all 4 of them).
From sequences to reals
sequence → ε-steady sequence → eventually ε-steady sequence → Cauchy sequence → ε-close sequences → eventually ε-close sequences → equivalent sequences → real numbers.Example
Let \( a_0, a_1, a_2, ... \) be the sequence:
\[ 1.4, 1.41, 1.414, 1.4142, 1.41421, ... \]
and let \( b_0, b_1, b_2, ... \) be the sequence:
\[ 1.5, 1.42, 1.415, 1.4143, 1.41422, ... \]
Then \( LIM_{n \rightarrow \infty}a_n \) is a real number and is the same real number as \( LIM_{n \rightarrow \infty}b_n \), as their corresponding Cauchy sequences are equivalent Cauchy sequences.