Convergent Sequences in Metric Spaces

Limits of Sequences in Metric Spaces

Recall that if a sequence of real numbers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_n = x} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \in \{ 1, 2, ... \}} , then the sequence:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad (x_n)_{n=1}^{\infty} = (x)_{n=1}^{\infty} = (x, x, ..., x, ...) \end{align}}

Furthermore, it's not hard to see that this sequence converges to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , i.e., Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} x_n = x} , i.e., Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} d(x_n, x) = 0} since for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_n} we have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x_n, x) = 0} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.