This is indeed true. See for example Lemma 2.60 in Kollar-Mori Birational Geometry of algebraic varieties. In particular, it is shown that a Cartier divisor $D$ is big if and only if $mD \sim A + E$ for some ample divisor $A$ and effective divisor $E$. This is also proven in Corollary 2.2.7 in Lazarsfeld's Positivity of in Algebraic Geometry I.
This is indeed true. See for example Lemma 2.60 in Kollar-Mori Birational Geometry of algebraic varieties. In particular, it is shown that a Cartier divisor $D$ is big if and only if $mD \sim A + E$ for some ample divisor $A$ and effective divisor $E$. This is also proven in Corollary 2.2.7 in Lazarsfeld's Positivity of in Algebraic Geometry I.