While walking in the park, I realozed that
The Binomial Theorem says
$$
\begin{align}
1+a_{n+1}
&=\left(1+2013a_n\right)^{1/2013}\\
&=1+a_n-1006a_n^2+O\left(a_n^3\right)\tag{1}
\end{align}
$$
Bernoulli's Theorem guarantees that $a_{n+1}\lt a_n$. Since $a_n\gt0$, $a_n$ must have a limit which satisfies
$$
(1+x)^{2013}=1+2013x\tag{2}
$$
Bernoulli's Theorem says that for $x\gt0$, the left side of $(2)$ is greater, so the only non-negative solution is $x=0$. Thus,
$$
\lim_{n\to\infty}a_n=0\tag{3}
$$