Complete Metric Spaces

Complete Metric Spaces

Recall that a sequence $(x_{n})_{n=1}^{\infty }$ in $M$ is called a Cauchy sequence if for all $\epsilon >0$ there exists an $N\in \mathbb {N}$ such that if $m,n\geq N$ then $d(x_{m},x_{n})<\epsilon$.

Consider any metric space $(M,d)$. If $(M,d)$ is such that every Cauchy sequence converges in $M$, then we give this metric space a special name.

Definition: Let $(M,d)$ be a metric space. Then $(M,d)$ is said to be Complete if every Cauchy sequence converges in $M$.

In general, it is much easier to show that a metric space is not complete by finding a Cauchy sequence that does not converge in the space. For example, consider the set $M=[0,1)$ with the standard Euclidean metric $d(x,y)=\mid x-y\mid$ defined 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, y \in M} . 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 (M, d)} is a metric space. Now, consider the following 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} :

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} = \left ( 1 - \frac{1}{n} \right )_{n=1}^{\infty} = \left ( 0, \frac{1}{2}, \frac{2}{3}, ... \right ) \end{align}}

We claim 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 (x_n)_{n=1}^{\infty}} is a Cauchy sequence. Let's prove this. 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, n \in \mathbb{N}} , and consider:

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_m, x_n) = \biggr \lvert \left ( 1 - \frac{1}{m} \right ) - \left ( 1 - \frac{1}{n} \right ) \biggr \rvert = \biggr \lvert \frac{1}{n} - \frac{1}{m} \biggr \rvert \leq \biggr \lvert \frac{1}{n} \biggr \rvert + \biggr \lvert \frac{1}{m} \biggr \rvert = \frac{1}{n} + \frac{1}{m} \end{align}}

Choose 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} such 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 > \frac{2}{\epsilon}} . Then 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, n \in \mathbb{N}} are such 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 m, 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 m \geq N > \frac{2}{\epsilon} \quad \mathrm{so} \quad \frac{1}{m} \leq \frac{1}{N} < \frac{\epsilon}{2} \\ \quad n \geq N > \frac{2}{\epsilon} \quad \mathrm{so} \quad \frac{1}{n} \leq \frac{1}{N} < \frac{\epsilon}{2} \end{align}}

Hence 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 m, n \geq 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 \begin{align} \quad d(x_m, x_n) \leq \frac{1}{n} + \frac{1}{m} < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon \end{align}}

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 (x_n)_{n=1}^{\infty}} is indeed a Cauchy sequence. However, it should be intuitively clear 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 \lim_{n \to \infty} x_n = 1} , but 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 1 \not \in [0, 1)} ! 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_n)_{n=1}^{\infty}} does not converge 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} and hence 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)} is not a complete metric space.

Licensing

Content obtained and/or adapted from:

• Complete Metric Spaces, mathonline.wikidot.com under a CC BY-SA license