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

ε-steadiness

Let \( \varepsilon > 0 \) be a rational. A sequence \( (a_n)_{n=m}^{\infty} \) is ε-steady iff each pair \( a_j, a_k \) of sequence elements are ε-close for every \( j,k \ge m \).

In other words, the sequence \( a_0, a_1, a_2, ... \) is ε-steady iff each \( |a_j - a_k| \le \varepsilon \) for all \( j,k \).



Example

The sequence \( 1, 0, 1, 0, 1,... \) is 1-steady, but not 0.5-steady. The sequence \( 0.1, 0.01, 0.001, 0.0001, ... \) is 0.1 steady, but not 0.01 steady. The sequence \( 1,2,3,4,5,... \) is not steady for any ε, and the sequence \( 2, 2, 2,... \) is steady for all \( \varepsilon > 0 \).

Context

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.

Source

Tao, Analysis I