Holomorphic function on unit disc with specific properties

Question

I am supposed to characterize all holomorphic functions on the unit disc on to the complex plane, such that there exists $N\in\mathbb{N}$ such that for all $n>N$: $\lvert f(\frac{1}{n})\rvert\leq e^{-n}$
My first attempt was to use identity principle but that wouldn't work since $exp(1/z)$ isn't even holomorphic on the unit disc. Is identity principle the right thought for this one? How can I find a function to use on that one?