Math and science::Analysis::Tao::05. The real numbers
Eventually ε-close sequences
Let \( (a_n)_{n=0}^{\infty} \) and \( (b_n)_{n=0}^{\infty} \) be two sequences of rational numbers and let \( \varepsilon >0 \) be a rational. The sequences are said to be eventually ε-close iff there exists an integer \( N \ge 0 \) such that the sequences \( (a_n)_{n=N}^{\infty} \) and \( (b_n)_{n=N}^{\infty} \) are ε-close.In other words \( a_0, a_1, a_2, ... \) is eventually ε-close to \( b_0, b_1, b_2, ... \) if there exists an \( N \ge 0 \) such that \( |a_j - b_j| \le \varepsilon \) for all \( j \ge N \).
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 not 0.1-close but are eventually 0.1-close. They are also eventually 0.01-close.