Show Question
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.

Tao, Analysis I