Math and science::Analysis::Tao::05. The real numbers
ε-close sequences
Let \( (a_n)_{n=0}^{\infty} \) and \( (b_n)_{n=0}^{\infty} \) be two sequences and let \( \varepsilon > 0 \) be a rational. \( (a_n)_{n=0}^{\infty} \) is said to be ε-close to \( (b_n)_{n=0}^{\infty} \) iff \( a_k \) is ε-close to \( b_k \) for all \( k \ge 0 \).In other words, the sequence \( a_0, a_1, a_2, ... \) is ε-close to the sequence \( b_0, b_1, b_2, ... \) iff \( |a_k - b_k| \le \varepsilon \) for all \( k = 0, 1, 2, ... \).
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:
\[ 1, -1, 1, -1, 1, ... \]
and
\[ 1.1, -1.1, 1.1, -1.1, 1.1, ... \]
are 0.1-close to each other. Note how neither of the sequences are 0.1-steady.