I’m trying to show that the following sequence of functions is or is not a Cauchy sequence on $C^0[-1,1]$ space under $\|.\|_{\infty}$ but I cannot obtain something concrete.
Consider the sequence of functions $(f_k)$ where each $f_k : [−1, 1] → \mathbb R$ is defined by:
$$f_k(x) = \begin{cases} -1, & \text{if $-1\le x\le -1/k$} \\ kx, & \text{if $-1/k\le x\le 1/k$} \\ 1, & \text{if $1/k\le x\le 1$} \end{cases}$$
Is $(f_k)$ a Cauchy sequence on $C^0_∞[−1, 1]$?
Should I use the negation of the Cauchy Sequence definition?