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

Eventually ε-close sequences

Let

(a_{n})_{n = 0}^{\infty}

and

(b_{n})_{n = 0}^{\infty}

be two sequences of rational numbers and let

ε > 0

be a rational. The sequences are said to be eventually ε-close iff there exists an integer

N \geq 0

such that the sequences

(a_{n})_{n = N}^{\infty}

and

(b_{n})_{n = N}^{\infty}

are ε-close.

In other words $a_{0}, a_{1}, a_{2}, . . .$ is eventually ε-close to $b_{0}, b_{1}, b_{2}, . . .$ if there exists an $N \geq 0$ such that $| a_{j} - b_{j} | \leq ε$ for all $j \geq N$ .

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

The two sequences

1.1, 1.01, 1.001, 1.0001, . . .

and

0.9, 0.99, 0.999, 0.9999, . . .

are not 0.1-close but are eventually 0.1-close. They are also eventually 0.01-close.

Source

Tao, Analysis I

23.08.2019