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 $\langle x_n \rangle$.

Examples
In metric spaces, or more generally, in Hausdorff spaces, if $x_n$ converges to a limit, then it can only have one subsequential limit which is the limit itself (see Limit of a Subsequence). However, if the sequence diverges or the space is not Hausdorff, there may be many different subsequential limits.

For instance, consider the series $0,1,0,1\dots$ that oscillates between $0$ in the odd numbers $x_{2n-1}$ and $1$ on the even numbers $x_{2n}$. The subsequence $\langle x_{n_r} \rangle$ where $n_r$ denotes the $r$th odd number has a subsequential limit of $0$ whereas the subsequence $\langle x_{n_r} \rangle$ where $n_r$ denotes the $r$th even number has a subsequential limit of $1$.