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

# Eventual ε-steadiness

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 \( \varepsilon > 0 \).