What can say about the behaviour of the sum $S$?

Question

First, let us consider $0<s<1$ and $0<\epsilon <1$ fixed. Suppose we have to cover $[0,1]$ by closed intervals each of whose length is less than $\epsilon$. Suppose we name those intervals $U=\{A_i^{\epsilon}\}_{i=1}^\infty$, hence, $l(A_i^{\epsilon})<\epsilon$ where $l(A_i^{\epsilon})$ is the length of the interval. Not only that, we have to cover it such a way that $S=\sum_{i=1}^\infty (l(A_i^{\epsilon}))^s$ is the least. What can say about the behaviour of the sum $S$? Will it increase if $\epsilon \to 0$?