Is the Heine Borel Theorem valid in $L^1$?

Question

I know that the Heine Borel Theorem is applicable to $R^n$. However, is the theorem still valid when we consider the $L^1$ space? Specifically, the $L^1$ with Lebesgue measure, where all functions in the space have a domain of [0,1].

I was given a hint to try a sequence where $||f_n||_1$ = 1, but $||f_n - f_m||_1 > c$, where c is a constant. However, I cannot think of any sequence in the $L^1$ space that fulfills this requirement.

Answer 1

Hint: try to find a sequence of nonnegative functions which have disjoint supports, yet all have integral 1 (i.e. area 1 under the graph). Draw some pictures.