Math and science::Analysis::Tao::06. Limits of sequences

# Cauchy sequences of reals

We can define Cauchy sequences of reals in the same way as was done for rationals. For rationals, many ideas of closeness were introduced separately before reaching Cauchy sequences. This card compresses all of the ideas for reals.

#### Distance and ε-close reals

Let $$a$$ and $$b$$ be two real numbers. Define their distance $$d(a,b)$$ to be  [...]. Let $$\varepsilon > 0$$ be a real. The two reals $$a$$ and $$b$$ are said to be ε-close if $$d(a,b) \le \varepsilon$$.

#### Sequence of reals

A sequences of reals is denoted in the same way as a sequences of rationals: $$(a_n)_{n=m}^{\infty}$$ represents a function $$f$$ from the set [...] to the set [...], where $$a_n := f(n)$$.

A sequence $$(a_n)_{n=N}^{\infty}$$ of reals is said to be ε-steady if $$a_j$$ and $$a_k$$ are [...] for all [...].
A sequence $$(a_n)_{n=m}^{\infty}$$ of reals is said to be eventually ε-steady if $$(a_n)_{n=N}^{\infty}$$ is [...] for some integer [...].
A sequence $$(a_n)_{n=m}^{\infty}$$ of reals is said to be a Cauchy sequence if it is [...] for every real $$\varepsilon > 0$$.