1.1 Completeness

Suppose that $M=(0,2)$ , i.e., the open interval from 0 to 2 on the real line, and let $d(x,y)=|x-y|$ . Consider the sequence defined by $x_i=\frac{1}{i}$ . $\left\{x_i\right\}$ is Cauchy since for any $\epsilon$ we can set $N$ such that $\frac{1}{N}<\frac{\epsilon }{2}$ , so that $|x_i-x_j|\le |x_i|+|x_j|<\frac{\epsilon }{2}+\frac{\epsilon }{2}=\epsilon$ . However, $x_i\to 0$ , but $0\notin M$ , i.e., the sequence converges to something that lives outside of our space.

Suppose that $M=C[-1,1]$ (the set of continuous functions defined on $[-1,1]$ ) and let $d_2$ denote the $L_2$ metric. Consider the sequence of functions defined by

• $M=[0,1],d(x,y)=|x-y|$ is complete.
• $(C[-1,1],d_2)$ is not complete, but one can check that $(C[-1,1],d_{\infty })$ is complete. (This space works because using $d_{\infty }$ , the above example is no longer Cauchy.)
• $\mathbb{Q}$ is not complete, but $\mathbb{R}$ is.