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

ε-close sequences

Let

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

and

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

be two sequences and let

ε > 0

be a rational.

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

is said to be ε-close to

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

iff

a_{k}

is ε-close to

b_{k}

for all

k \geq 0

In other words, the sequence $a_{0}, a_{1}, a_{2}, . . .$ is ε-close to the sequence $b_{0}, b_{1}, b_{2}, . . .$ iff $| a_{k} - b_{k} | \leq ε$ for all $k = 0, 1, 2, . . .$ .

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, - 1, 1, . . .

and

1.1, - 1.1, 1.1, - 1.1, 1.1, . . .

are 0.1-close to each other. Note how neither of the sequences are 0.1-steady.

Source

Tao, Analysis I

23.08.2019