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.