How to prove that cake-eating problem has no solution?

Question

Consider the following optimization problem (called cake-eating):

$$\sum\limits_{t = 0}^{\infty} u(a_t) \to \max$$

subject to

$$\sum\limits_{t = 0}^{\infty} a_t \leq s, \quad s >0, \quad a_t \geq 0$$

Show that if $u(a)$ is increasing, whose derivative at the zero point tends to infinity, and $u(a)$ being a strictly concave then this problem has no solution.

Answer 1

Hint: Essentially, if the utility function is as described in your setting, you can increase the utility by spreading out you cake eating further into smaller chunks. If the derivative of $u$ tends to infinity at zero then for sufficiently small $a$ you have $2u(a/2) > u(a)$. So for any given sequence $a_t$ you can improve utility by cutting your daily slice in half and eating it over two days.