Ampleness is invariant under positive powers

Tensor powers and sections. By [F1] the sheaf $L^m$ is invertible, and the quasi-compactness clause in [F2] concerns only $X$, so it holds for $L$ and for $L^m$ simultaneously. For every $n\ge 1$ there is a canonical identification $(L^m{)}^n=L^{mn}=L^{nm}=(L^n{)}^m$, and accordingly a section $s\in \mathrm{\Gamma }(X,L^n)$ of a power of $L$ has an $m$-th tensor power $s^m\in \mathrm{\Gamma }(X,L^{nm})=\mathrm{\Gamma }(X,(L^m{)}^n)$.

The nonvanishing locus is unchanged. For $x\in X$ let $\bar{s}$ denote the image of $s$ in the one-dimensional $\kappa (x)$-vector space $L^n{\otimes }_{{\mathcal{O}}_X}\kappa (x)$; the image of $s^m$ in $(L^n{)}^{\otimes m}\otimes \kappa (x)=(L^n\otimes \kappa (x){)}^{\otimes m}$ is ${\bar{s}}^{\otimes m}$, so by [F3] the point $x$ lies in $X_{s^m}$ if and only if it lies in $X_s$. Hence $X_{s^m}=X_s$ for every $s$ and every $m\ge 1$.

Ample implies power ample. Assume $L$ ample and let $x\in X$. By [F2] there are $n\ge 1$ and $s\in \mathrm{\Gamma }(X,L^n)$ with $x\in X_s$ and $X_s$ affine. By step 1.1 the section $s^m$ lies in $\mathrm{\Gamma }(X,(L^m{)}^n)$, and by step 2.1 its nonvanishing locus $X_{s^m}$ equals the affine open $X_s$ containing $x$. Hence $L^m$ is ample by [F2].

Power ample implies ample. Assume $L^m$ ample and let $x\in X$. By [F2] applied to the invertible sheaf $L^m$ there are $n\ge 1$ and $t\in \mathrm{\Gamma }(X,(L^m{)}^n)$ with $x\in X_t$ and $X_t$ affine. Interpreting $t$ as a section of $L^{mn}=(L^m{)}^n$ by the identification of step 1.1, the nonvanishing locus computed in the line bundle $L^{mn}$ is the same set $X_t$, affine and containing $x$; since $mn\ge 1$, this witnesses ampleness of $L$ by [F2].

Boundaries and conclusion. If $X=\mathrm{\varnothing }$ both conditions hold vacuously by [F2]. If $m=1$ the two conditions coincide, and steps 3.1 and 3.2 prove the two implications for arbitrary $m\ge 1$. [F2, step 3.1, step 3.2] \qed