Now that makes more sense and it would've made my "proof" valid.
So we have y(t) > 0, increasing and unbounded. For the function not to diverge (go to infinity), it would have to converge to a real value or oscillate. It cannot oscillate since it's increasing, y(b) > y(a) for every b > a. It can't converge to a real value (say L) either, since that would make it bounded (by L).