Complete metric space

Definition

Given a metric space $(X,d)$, if every Cauchy sequence converges to a limit (sequence) within $X$ then $X$ is a complete metric space. That is to say:

• Given a sequence $(x_n)_{n=1}^\infty$, it converging to a limit $x\in X$ or being a Cauchy sequence are equivalent. Or in symbols:
• $\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n>N\implies d(x_n,x)]\iff\forall\epsilon>0\exists N\in\mathbb{N}\forall n,m\in\mathbb{N}[n\ge m>N\implies d(x_n,x_m)<\epsilon]$

Examples

Complete space

• Obviously the $\mathbb{R}$ (reals) are complete, considered with the usual topology induced by the Absolute value metric

Incomplete space

• A good example is the space of fractions, $\mathbb{Q}$ considered with the Absolute value metric again, there are rational sequences which converge to say, $\sqrt{2}$, and $\sqrt{2}\notin\mathbb{Q}$
• A better example is the space of continuous functions on an interval, $\mathcal{C}[a,b]$ and the distance function:
  • $d(f,g)=\sqrt{\int^b_a\vert f(x)-g(x)\vert dx}$ for $f,g\in\mathcal{C}[a,b]$
  • Let $a=-1$ and $b=1$ (WLOG)
  • We can then see that the sequence of functions $(f_n)_{n=1}^\infty$ where each $f_n:[-1,1]\rightarrow[0,1]\subset\mathbb{R}$ given by:

    $f_n(x)=\left\{\begin{array}{lr}0 & \text{for }x\in[-1,0] \\ nx &\text{for }x\in(0,\frac{1}{n}] \\ 1 & \text{otherwise}\end{array}\right.$

    • Has a limit (note that: $\lim_{n\rightarrow\infty}(f_n)=f$ with $f(x)=\left\{\begin{array}{lr} 0 & \text{for }x\in[-1,0] \\ 1 & \text{otherwise}\end{array}\right.$ and that this $f$ isn't continuous (in $(\mathbb{R},\vert\cdot\vert)$) anyway!)
    • and that limit, $f$ isn't continuous, this we have shown that $\mathcal{C}[-1,1]$ isn't complete. (and by translation/scaling as needed, $\mathcal{C}[a,b]$ isn't complete)