Math and science::Analysis::Tao::05. The real numbers

In other words, a sequence $(a{)}_{n=m}^{\mathrm{\infty }}$ is eventually ε-steady iff there exists an integer $N\ge m$  such that $|a_j-a_k|\le \epsilon$ for all $j,k\ge N$.

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 sequence $1,0.1,0.001,...$ is not 0.1-steady, but is eventually 0.1-steady. The sequence $10,0,0,0,...$ is not ε-steady for any ε less than 10, but it is eventually ε-steady for every $\epsilon >0$.

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Source

Tao, Analysis I