Subsequential Limit

Contents

Definition

Definition

Let $(s_n)$ be a bounded sequence (not necessarily convergent), and let $(s_{n_i})$ be a convergent subsequence of $(s_n)$ . Then, if $s$ is the limit of $(s_{n_i})$ , we call $s$ a subsequential limit of $(s_n)$ .

Theorem 1 – If $(s_n)$ is convergent with limit $L$ , then $lim (s_{n_i}) = L$ for every subsequence $(s_{n+1})$ .

Proof – See here.