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

ε-steadiness

Let

ε > 0

be a rational. A sequence

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

is ε-steady iff each pair

a_{j}, a_{k}

of sequence elements are ε-close for every

j, k \geq m

In other words, the sequence $a_{0}, a_{1}, a_{2}, . . .$ is ε-steady iff each $| a_{j} - a_{k} | \leq ε$ for all $j, k$ .

Example

The sequence $1, 0, 1, 0, 1, . . .$ is 1-steady, but not 0.5-steady. The sequence $0.1, 0.01, 0.001, 0.0001, . . .$ is 0.1 steady, but not 0.01 steady. The sequence $1, 2, 3, 4, 5, . . .$ is not steady for any ε, and the sequence $2, 2, 2, . . .$ is steady for all $ε > 0$ .

Context

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.

Source

Tao, Analysis I

22.08.2019