I know that $(C[0, 2], d_{\infty})$ is a complete metric space, being $C[0, 2]$ the set of continuous functions in the closed interval $[0, 2]$ and $d_\infty$ the distance metric induced by the infinite norm, i.e., $d_\infty\{f(x), g(x)\} = ||f(x) - g(x)||_\infty = \sup \{|f(x) - g(x)|\}$ with $x \in C[0, 2]$.
Given the following sequence of functions:
$f_n(x) = \begin{cases} 0 & \mbox{if } 0 \leq x \leq 1 - \frac{1}{n}, \\ nx + 1 - n & \mbox{if } 1 - \frac{1}{n} < x \leq 1, \\ 1 & \mbox{if } 1 < x \leq 2, \end{cases}$
I would like know if, in this metric space, this sequence is a Cauchy sequence, and if so, to which limit it converges.
The origin of this question is that, using a norm-2 induced metric, this sequence converges to the non continuous step function. This shows that $(C[a, b], d_{2})$ is not complete. But I when I tried to calculate the limit of this sequence for $d_\infty$ I found myself lost.
I would really appreciate any hint.
Thanks in advance!