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.