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 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 $1$.