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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 $$\varepsilon >0$$ be a rational. The sequences are said to be eventually ε-close iff there exists an integer $$N \ge 0$$ 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 $$N \ge 0$$ such that $$|a_j - b_j| \le \varepsilon$$ for all $$j \ge N$$.

#### 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.

Tao, Analysis I