Since $f_{L_{i}}L_{i}$ converges to $f_{L}L$ as currents,

It suffices to justify $\int_{\mathcal{C}_{i}}\lambda\wedge\Omega\to\int_{\mathcal{C}}\lambda\wedge\Omega,$ where $\partial\mathcal{C}_{i}=L_{i}-L_{0}$, and $\partial\mathcal{C}=L-L_{0}$.

Now the flat norm convergence gives $L_{i}-L=A_{i}+\partial B_{i}$ for some integral currents $A_{i},B_{i}$, with $Mass(A_{i})+Mass(B_{i})\to 0$. Since $[L_{i}]=[L]=[L_{0}]\in H_{n}(X)$, the homology class of $A_{i}$ is zero. A version of the isoperimetric theorem (cf. Prop. 5.11 below) then says $A_{i}=\partial B_{i}^{\prime}$ with $Mass(B_{i}^{\prime})\leq CMass(A_{i})^{(n+1)/n}\to 0$. Without loss of generality we absorb $B_{i}^{\prime}$ into $B_{i}$. Then we can simply choose $\mathcal{C}_{i}=\mathcal{C}+B_{i}$, which is legitimate since it satisfies $\partial\mathcal{C}_{i}=L_{i}-L_{0}$. The claim follows by

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