We see that the pointwise limit is $f(x)=1$ if $x=1$, and $f(x)=0$, for $x\in[0,1)$. (our background space is $C[0,1]$)
How would one go about proving if $f_n\to f$? My feeling is that we do not have convergence, since the limit is not continuous.
But since $f=1$ at only one point, would it be fair to argue:
$d_1(f_n,f)=\int_0^1|f_n-f|dx=\int_0^1x^ndx=\frac{1}{n+1}\to 0$ as $n\to\infty$