Contractive sequence

Definition

Let $(a_n)$ be a real sequence. The sequence is contractive if there is a real number $C$ with $0 < C < 1$ such that $$|a_{n+2} - a_{n+1}| \le C\, |a_{n+1} - a_n| \quad \text{for all } n \in \mathbb{N}.$$

Remarks

Each contractive sequence is a Cauchy sequence. By induction $|a_{n+1} - a_n| \le C^n |a_1 - a_0|$, and the geometric series bounds $|a_m - a_n|$. So each contractive sequence converges. If $f \colon \mathbb{R} \to \mathbb{R}$ is a contraction mapping, so that $|f(x) - f(y)| \le C|x - y|$ with $C < 1$, then each sequence $a_{n+1} = f(a_n)$ is contractive. It is not enough that $|a_{n+2} - a_{n+1}| < |a_{n+1} - a_n|$ for all $n$. The partial sums of the harmonic series satisfy this, and they diverge.

Explanations

L1 human

In a contractive sequence, each step is at most a fixed fraction of the step before. The steps shrink like a geometric sequence, so the total distance travelled is finite, and the sequence converges.