Convergent Sequences in Metric Spaces

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:

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

Furthermore, it's not hard to see that this sequence converges to $x$, 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.

The Boundedness of Convergent Sequences in Metric Spaces

If $(M,d)$ is a metric space and $(x_{n})_{n=1}^{\infty }$ is a sequence in $M$ that is convergent then the limit of this sequence $p\in M$ is unique.

We will now look at another rather nice theorem which states that if $(x_{n})_{n=1}^{\infty }$ is convergent then it is also bounded.

Theorem 1: Let $(M,d)$ be a metric space and let $(x_{n})_{n=1}^{\infty }$ be a sequence in $M$. If $(x_{n})_{n=1}^{\infty }$ is convergent then the set $\{x_{1},x_{2},...,x_{n},...\}$ is bounded.

• Proof: Let $(M,d)$ be a metric space and let $(x_{n})_{n=1}^{\infty }$ 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 $M=\max\{M^{*},1\}$. Then for all $n\in \mathbb {N}$, $d(x_{n},p)\leq M$. So consider the open ball $B(p,M+1)$. Then $x_{n}\in B(p,M+1)$ for all $n\in \{1,2,...\}$ so:

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

• Therefore $\{x_{1},x_{2},...,x_{n},...\}$ is a bounded set in $M$. $\blacksquare$

Convergent Sequences and Subsequences in Metric Spaces

We will now look at some nice criterion which tells us that in a metric space $(M,d)$, a sequence $(x_{n})_{n=1}^{\infty }$ converges to $p$ if and only if every subsequence $(x_{n_{k}})_{k=1}^{\infty }$ converges to $p$. This is analogous to the similar result when we looked at convergent sequences of real numbers.

Theorem 1: Let $(M,d)$ be a metric space, let $(x_{n})_{n=1}^{\infty }$ be a sequence in $M$, and let $p\in M$. Then $(x_{n})_{n=1}^{\infty }$ converges to $p$ if and only if every subsequence $(x_{n_{k}})_{k=1}^{\infty }$ of $(x_{n})_{n=1}^{\infty }$ converges to $p$.

• Proof: $\Rightarrow$ Let $(x_{n})_{n=1}^{\infty }$ be a convergent sequence in $M$ that converges to $p$. 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 \lim_{n \to \infty} x_n = p} . So $\lim _{n\to \infty }d(x_{n},p)=0$. Therefore, for all $\epsilon >0$ there exists an $N=N(\epsilon )\in \mathbb {N}$ such 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 n \geq N} 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 \begin{align} \quad d(x_n, p) < \epsilon \end{align}}

• 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_k})_{k=1}^{\infty}} be any subsequence of 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}} . For each 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 \epsilon > 0} we have that for some 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 K \in \mathbb{N}} 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 n_K \geq N(\epsilon)} and so 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 k \geq K} 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 n_k \geq N(\epsilon)} so by the convergence of 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}} 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 \begin{align} \quad d(x_{n_k}, p) < \epsilon \end{align}}

• So for each 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 \epsilon > 0} there exists a 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 K \in \mathbb{N}} such 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 k \geq K} 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 d(x_{n_k}, p) < \epsilon} , therefore, the subsequence 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_k})_{k=1}^{\infty}} 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 p} .

• 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 \Leftarrow} Suppose that every subsequence 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_k})_{k=1}^{\infty}} of 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}} 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 p} . 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)_{n=1}^{\infty}} is a subsequence of itself and 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 p} . 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
• Convergent Sequences and Subsequences in Metric Spaces, mathonline.wikidot.com under a CC BY-SA license