Math and science::Analysis::Tao::05. The real numbers
Sequences
Let \( m \) be an integer. A sequence \( (a_n)_{n=m}^{\infty} \) of rational numbers is any function from the set \( \{ n \in \mathbb{Z} : n \ge m \} \) to \( \mathbb{Q} \).In other words, a sequence is a mapping that assigns to each integer greater than or equal to \( m \) a rational number \( a_n \).
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
\( (n^2)_{n=0}^{\infty} \) is the collection 0, 1, 4, 9..., where the function \( f: \mathbb{Z} \rightarrow \mathbb{Q} \) is the set of tuples \( \{ (0, 0), (1, 1), (2,4), (3, 9)... \} \) where for every first element of the tuple, \( n \), the second element is \( n^2 \).