Is $\limsup_{t} \frac{a_t}{ h(t+1) - h(t) }$ finite when $0 \leq \sum_{i=t}^\infty a_i \leq h(t) = t^{-\alpha}$?

You see for all $t$,by monotonicity of $a$, we have: $$ t^{\alpha+1} a_{2t} \le t^{\alpha} \sum_{n \ge t+1} a_n \le 1$$ Hence, $$ \limsup t^{\alpha+1} a_{t} \le 2^{\alpha+1}$$ Q.E.D