It may not be as stated as such in Reider's paper (I don't have it at hand), but it is an easy consequence of his results : Reider proves that if $N$ is nef and $N^2\geq 10$, $K_X+N$ is very ample unless there is an effective divisor $E$ on $X$ with $E.N\leq 2$ (+ some extra conditions). If $N=4L$ with $L$ ample, clearly such a divisor cannot exist, and $N^2\geq 16$, q.e.d.

It is not stated as such in Reider's paper, but it is an easy consequence of his main theorem (thm. 1) : Reider proves that if $N$ is nef and $N^2\geq 10$, $K_X+N$ is very ample unless there is an effective divisor $E$ on $X$ with $E.N\leq 2$ (+ some extra conditions). If $N=4L\ $ with $L$ ample, clearly such a divisor cannot exist, and $\ N^2\geq 16$, q.e.d.