Finite limit and convergent sequences

The central idea of the chapter is that of the limit: towards which number do the terms of a sequence “accumulate”.

Definition — Finite limit

The sequence $(a_n)$ is said to have limit $L\in\mathbb{R}$ (and one writes $\lim_{n\to+\infty}a_n = L$, or more briefly $a_n\to L$) if for every $\varepsilon>0$ there exists an index $n_0\in\mathbb{N}$ such that $|a_n - L|<\varepsilon$ for every $n\ge n_0$.

A sequence that admits a finite limit is said to be convergent.

Observation — Reading

“However small an $\varepsilon$ you choose, from a certain $n_0$ onwards all the terms are at a distance less than $\varepsilon$ from $L$.” The terms accumulate around $L$, leaving out at most a finite number of them.

Links

Topics: Sequences
Concepts: Limit of a sequence · Convergent sequence
Methods: Limit of a sequence
Skills: Compute limits