I am studying "real analysis and applications :theory in practice" I have some question in this books'problem. so if you know following ploblem, please teach me ! the ploblem is as follows
excercises for section 7.2 I with N
consider the piecewise linear fuctions in $C[-1,1]$ given by $f_n(x)=0$ for $-1\leq x\leq 0$, $f_n(x)=nx$ for $0\leq x\leq \frac1n$, and $f_n(x)=1$ for $\frac1n\leq x\leq 1$ with $L^1[-1,1]$ norm
(a) show that $f_n$ is Cauchy in the $L^1$-norm (already clear for your help)
(b) show that $f_n$ converges $\chi_\left(0,1\right]$, the characteric function of (0,1], in the $L^1-norm$
(c) show that $\Vert \chi_\left(0,1\right]-h\Vert_1 \gt 0$ for every h in C[-1,1]
I solved above 3 question but problem is as follows
(d) conclude that C[-1,1] is not complete in the $L^1-norm$
this is what i was trying to solve.
(1), what we need to prove this problem suffice to show there is cauchy sequence whose limit is not in C[-1,1].
(2)then suppose (a),(b) is true, (a),(b) said if we choose piecewise function like above fnfn. it converges to a vector $\chi_\left(0,1\right]$ but $\chi_\left(0,1\right]$ is not in C[-1,1] because $\chi_\left(0,1\right]$ is not continuous function on [-1,1].
(3)thus C[-1,1] is not complete.
is there any faults without using problem (c)?