In Example 2.2.19 of

Lazarsfeld, Positivity in Algebraic Geometry I,

I found the following statement:

Let $D$ be a divisor on an irreducible projective variety $X$. Then $D$ is nef and big if and only if there exists an effective divisor $N$ such $D-\frac{1}{k}N$ is ample for $k\gg 0$.

It is clear to me that $D$ nef and big implies that there exists an effective divisor $N$ such $D-\frac{1}{k}N$ is ample for $k\gg 0$. It is quite clear that $D-\frac{1}{k}N$ ample implies $D$ big.

I can not understand why $D-\frac{1}{k}N$ ample implies $D$ nef. Perhaps I am just misunderstanding the staments. We may write $\epsilon = \frac{1}{k}$ and $D-\epsilon N$. Does the statement mean $D-\epsilon N$ is ample for some $\epsilon$ small enough or for any $0< \epsilon\ll 1$ ?