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

Let $$\varepsilon > 0$$ be a rational. A sequence $$(a)_{n=m}^{\infty}$$ is eventually ε-steady iff the sequence $$a_N, a_{N+1}, a_{N+2}, ...$$ is ε-steady for some integer $$N \ge m$$

In other words, a sequence $$(a)_{n=m}^{\infty}$$ is eventually ε-steady iff there exists an integer $$N \ge m$$  such that $$|a_j - a_k| \le \varepsilon$$ for all $$j, k \ge N$$.

#### 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 sequence $$1, 0.1, 0.001, ...$$ is not 0.1-steady, but is eventually 0.1-steady. The sequence $$10, 0, 0, 0, ...$$ is not ε-steady for any ε less than 10, but it is eventually ε-steady for every $$\varepsilon > 0$$.

Tao, Analysis I