Limits of Sequences in Metric Spaces
Recall that if a sequence of real numbers $(x_n)_{n=1}^{\infty} = (x_1, x_2, ..., x_n, ...)$ is an infinite ordered list where $x_k \in \mathbb{R}$ for every $k \in \{ 1, 2, ... \}$. We will now generalize the concept of a sequence to contain elements from a metric space $(M, d)$.
| Definition: Let $(M, d)$ be a metric space. An (infinite) Sequence in $M$ denoted $(x_n)_{n=1}^{\infty} = (x_1, x_2, ..., x_n, ...)$ is an infinite ordered list of elements $x_k \in M$ for all $k \in \{1, 2, ... \}$. |
Finite sequences in a metric space can be defined as a finite ordered list of elements in $M$ but their study is not that interesting to us.
We can also define whether a sequence $(x_n)_{n=1}^{\infty}$ of elements from a metric space $(M, d)$ converges or diverges.
| Definition: Let $(M, d)$ be a metric space. A sequence $(x_n)_{n=1}^{\infty}$ in $M$ is said to be Convergent to the element $p \in M$ written $\lim_{n \to \infty} x_n = p$ if $\lim_{n \to \infty} d(x_n, p) = 0$ and the element $p$ is said to be the Limit of the sequence $(x_n)_{n=1}^{\infty}$. If no such $p \in M$ exists, then $(x_n)_{n=1}^{\infty}$ is said to be Divergent. |
There is a subtle but important point to make. In the definition above, $\lim_{n \to \infty} x_n = p$ represents the limit of a sequence of elements from the metric space $(M,d)$ to an element $p \in M$ while $\lim_{n \to \infty} d(x_n, p) = 0$ represents the limit of a sequence of positive real numbers to $0$ - such limits we already have experience with.
For example, if $M$ is any nonempty set, $d : M \times M \to [0, \infty)$ is the discrete metric, and $x \in M$, then the sequence defined by $x_n = x$ for all $n \in \{ 1, 2, ... \}$, then the sequence:
(1)Furthermore, it's not hard to see that this sequence converges to $x$, i.e., $\lim_{n \to \infty} x_n = x$, i.e., $\lim_{n \to \infty} d(x_n, x) = 0$ since for all $x_n$ we have that $d(x_n, x) = 0$, so $\lim_{n \to \infty} d(x_n, x) = \lim_{n \to \infty} 0 = 0$.
We will soon see that many of theorems regarding limits of sequences of real numbers are analogous to limits of sequences of elements from metric spaces.