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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 $$\varepsilon \gt 0$$, 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 $$\varepsilon \ge 0$$ there exists an $$N \ge 0$$ such that $$|a_k - b_k| \le \varepsilon$$ for all $$k \ge N$$.

#### 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 equivalent.

Tao, Analysis I