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 Z : n \geq m}

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 $f : Z \to Q$ 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 $n^{2}$ .

Source

Tao, Analysis I

22.08.2019