Metric Limits

Why

Once we have a notion of distance, we can define a more general notion of convergence.

Definition

Let $(X, d)$ be a metric space and let $(x_n)_{n \in \N }$ be a sequence in $X$. A limit of $(x_n)_{n \in \N }$ is an element $x \in X$ for which $d(x, x_n) \to 0$.