Let $X$ be a projective Calabi-Yau $n-$fold, and $\alpha\in N^{1}(X)_{\mathbb{R}}$ a big and nef class that is not ample. Then there exists $\omega\in\alpha$ a smooth real $(1,1)$ form that is pointwise nonnegative. Moreover if $\alpha_{t}:[0,1]\to\overline{\mathcal{K}}_{NS}$ is a smooth path such that $\alpha_{t}\in\mathcal{K}_{NS}$ for $t<1$ and $\alpha_{1}=\alpha$, then we can find a continuous family of Kähler forms $\beta_{t}\in\alpha_{t}$, $t<1$, such that $\beta_{t}\to\omega$ in the $C^{\infty}$ topology as $t$ approaches $1$.