Why does this sum converge to $0$?

Question

In Van der Poorten's "A Proof that Euler Missed...", which outlines Apéry's proof that $\zeta(3)$ is irrational, the following sum appears: $$ S=\sum_{k=1}^{N} \frac{(-1)^k}{(2k^3) \binom{N+k}{k} \binom {N}{k}}. $$ Van der Poorten remarks further on that $\displaystyle \lim_{N\to\infty} S = 0$ without giving a proof. This does not seem immediately obvious to me, how can I prove it?

Answer 1

Every term is less than $1/N^2$, so the total is less than $1/N$