Definition:Subsequential Limit

Definition
Let $\langle x_n \rangle$ be a sequence.

Let $\langle x_{n_r} \rangle$ be a subsequence of $\langle x_n \rangle$.

Suppose that $\langle x_{n_r} \rangle$ converges to a limit $x$.

Then $x$ is called a subsequential limit of $\langle x_n \rangle$.

Also see

 * First Subsequence Rule
 * Limit of Subsequence equals Limit of Sequence
 * Divergent Sequences may be Bounded