Suppose $L$ is an effective divisor and $H$ is ample (On a smooth 3-fold) such that $L+H$ is nef.


Then show that $L+H$ is big ( $(L+H)^3 > 0$) ?


This was claimed in a paper, without proof. So I assume it should be well-known.
I am not sure if the restriction on dimension is necessary or not.



Question on nef and big divisors
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*. 


Anyway, they make no assumptions on the dimension or singularities of the ambient variety, you also don't need the nef assumption.


