example of a sequence in $C[0,1]$ which converges in weighted sup norm but not in L1 norm

Question

I am trying to think of a sequence $\lbrace f_n \rbrace$ in $C[0,1]$ which converges in the weighted supremum norm $$\|f\| = \sup_{x \in [0,1]}|xf(x)|$$ but not in the L1 norm $$\|f\|_1 = \int_0^1 |f(x)|\ dx.$$

How do I construct such a sequence?