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

Equivalent sequences

Two sequences

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

and

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

are equivalent iff for each rational

ε > 0

, the sequences

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

and

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

are eventually ε-close.

In other words the sequences $a_{0}, a_{1}, a_{2}, . . .$ and $b_{0}, b_{1}, b_{2}, . . .$ are equivalent iff for every rational $ε \geq 0$ there exists an $N \geq 0$ such that $| a_{k} - b_{k} | \leq ε$ for all $k \geq N$ .

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 equivalent.

Source

Tao, Analysis I

23.08.2019