Let $X$ denote the set of all the real (or complex) valued continuous functions on the closed interval $[0, 1]$, and let $$ d(x, y) \colon= \int_0^1 \lvert x(t) - y(t) \rvert \ \mathrm{d} t $$ for all $x, y \in X$. This $d$ is a metric on $X$, and $(X, d)$ is not complete, as has been shown by Kreyszig, for the sequence $\left( x_n \right)$, where $$ x_n (t) \colon= \begin{cases} 0 \ & \mbox{ if } \ 0 \leq t \leq \frac{1}{2}, \\ (m+2)\left( t - \frac{1}{2} \right) \ & \mbox{ if } \ \frac{1}{2} \leq t \leq \frac{1}{2} + \frac{1}{m+2}, \\ 1 \ & \mbox{ if } \ \frac{1}{2} + \frac{1}{m+2} \leq t \leq 1, \end{cases} $$ for $n \in \mathbb{N}$, is a Cauchy sequence that fails to converge to any point $x$ in $(X, d)$.
Now my question is, what is the gist of the process involved in the construction of such examples? I mean what do we need to look for in the functions $x_n$ that would constitute a Cauchy but not a convergent sequence?
Another example given by Kreyszig is in Prob. 13, Sec. 1.5. Here $x_n$ is defined as $$ x_n(t) \colon= \begin{cases} n \ & \mbox{ if } \ 0 \leq t \leq n^{-2}, \\ t^{-1/2} \ & \mbox{ if } \ n^{-2} \leq t \leq 1. \end{cases} $$ What is the gist of examples like this one?
How do we generalise each one of these two examples to an arbitrary closed interval $[a, b]$, where $a$ and $b$ are some real numbers such that $a < b$?
The completion of this space under the given $L^1[0,1]$ metric is all of $L^1[0,1]$, which is just about as classical as it gets. So if you construct a sequence $\{ f_n \}$ in $C[0,1]$ that converges in this metric to some $f \in L^1[0,1]$ which is not equal a.e. to some $\tilde{f} \in C[0,1]$, then $\{ f_n \}$ will be Cauchy in the metric, but it will not be able to converge to any $g\in C[0,1]$ in this metric because of uniqueness of $L^1$ limits. So that's the intuition, and it's a proof if you know such facts about $L^1[0,1]$.