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 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_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 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_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 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 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 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 (M, d)} converges or diverges.

Definition: Let $(M,d)$ be a metric space. A sequence $(x_{n})_{n=1}^{\infty }$ in 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 M} is said to be Convergent to the element $p\in M$ written 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 = p} if $\lim _{n\to \infty }d(x_{n},p)=0$ and the element 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 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, 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 = p} represents the limit of a sequence of elements from the metric space $(M,d)$ to an element 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 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 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 M} is any nonempty set, 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 : 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:

${\begin{aligned}\quad (x_{n})_{n=1}^{\infty }=(x)_{n=1}^{\infty }=(x,x,...,x,...)\end{aligned}}$

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 $d(x_{n},x)=0$, 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.

The Boundedness of Convergent Sequences in Metric Spaces

If $(M,d)$ is a metric space and 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}} is a sequence in 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 M} that is convergent then the limit of this 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 p \in M} is unique.

We will now look at another rather nice theorem which states that if 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}} is convergent then it is also bounded.

Theorem 1: Let $(M,d)$ be a metric space and let 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}} be a sequence in $M$. If $(x_{n})_{n=1}^{\infty }$ is convergent then the set 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_1, x_2, ..., x_n, ... \}} is bounded.

• Proof: Let $(M,d)$ be a metric space and let 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}} be a sequence in $M$ that converges to $p\in M$, i.e., $\lim _{n\to \infty }x_{n}=p$. Then $\lim _{n\to \infty }d(x_{n},p)=0$. So for all $\epsilon >0$ there exists an $N\in \mathbb {N}$ such that if $n\geq N$ then $d(x_{n},p)<\epsilon$. So for $\epsilon _{0}=1>0$ there exists an $N(\epsilon _{0})\in \mathbb {N}$ such that if $n\geq N(\epsilon _{0})$ then:

${\begin{aligned}\quad d(x_{n},p)<\epsilon _{0}=1\end{aligned}}$

• Now consider the elements $x_{1},x_{2},...,x_{N(\epsilon _{0})}$. This is a finite set of elements and furthermore the set of distances from these elements to $p$ is finite:

${\begin{aligned}\quad \{d(x_{1},p),d(x_{2},p),...,d(x_{N(\epsilon _{0})-1},p)\}\end{aligned}}$

• Define $M^{*}$ to be the maximum of these distances:

${\begin{aligned}\quad M^{*}=\max\{d(x_{1},p),d(x_{2},p),...,d(x_{N(\epsilon _{0})-1},p)\}\end{aligned}}$

• So if $1\leq n<N(\epsilon _{0})$ we have that $d(x_{n},p)\leq M^{*}$ and if $n\geq N(\epsilon _{0})$ then $d(x_{n},p)<1$. Let 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 M = \max \{ M^*, 1 \}} . Then 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 \mathbb{N}} , 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, p) \leq M} . So consider the open ball 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 B(p, M + 1)} . Then 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 \in B(p, M + 1)} 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, ... \}} so:

${\begin{aligned}\quad \{x_{1},x_{2},...,x_{n},...\}\subseteq B(p,M+1)\end{aligned}}$

• Therefore 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_1, x_2, ..., x_n, ... \}} is a bounded set in 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 M} . 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 \blacksquare}

Licensing

Content obtained and/or adapted from:

• Limits of Sequences in Metric Spaces, mathonline.wikidot.com under a CC BY-SA license
• The Boundedness of Convergent Sequences in Metric Spaces, mathonline.wikidot.com under a CC BY-SA license