What is the definition for Cauchy sequence in $\mathcal{C}\,[0,1]$? I think the definition should be as following: for example, $f_n \in \mathcal{C}\,[0,1]$ is said to be Cauchy if for every positive real number $\varepsilon \gt 0$ there is a positive integer $N$ such that for all positive integers $m,n \gt N$, the distance
$\left|f_n(x) - f_m(x)\right|\lt\varepsilon$ while $x\in[0,1]$. Is that right?

Recall the topology on $C([0,1])$ is that of uniform convergence. Thus the condition you're looking for is that the sequence of functions be uniformly Cauchy: given $\epsilon >0,$ there is a positive integer $N$ such that for all positive integers $m,n > N,$ one has $\sup_{x \in [0,1]} |f_m(x) - f_n(x)| < \epsilon.$ 
