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.
