Complete in $L^1$ but not $\mathrm{sup}$ norm

Question

While learning a bit of functional analysis from an introductory book I got stumped by the following problem:

Find a linear space complete in the $L^1$ norm, $||f||_1 \equiv \int_0^1 |f(t)|\ \mathrm{d}t$, but not in the supremum norm.

I'm having trouble translating $L^1$ convergence to a useful property, unlike its supremum counterpart which gives uniform convergence. Thanks for any hints!