I am trying to understand the paper by Bombieri and Giusti on Harnack inequality on minimal surfaces: https://link.springer.com/article/10.1007/BF01418640.
In particular, I am trying to understand the proof of a relative isoperimetric inequality:
Theorem 2. Let $S$ be an oriented boundary of least area, in the open $(n+1)$-ball $B_r$ and assume $\partial S=0$. Let $S_i\in R_n^{loc}(B_r)$ $i=1,2$ satisfy $$S=S_1+S_2\quad \|S\|=\|S_1\|+\|S_2\|.$$ Then we have $$M(\partial S_1)=M(\partial S_2)\ge \frac{1}{2\gamma}\min\{M(S_1\llcorner B_{\beta r},M(S_2\llcorner B_{\beta r})\}^{\frac{n-1}{n}}\quad {(*)}$$ where $\beta>0$ depends only on $n$ and $\gamma$ is the usual isoperimetric constant: $M(T)^{\frac{n-1}{n}}\le \gamma M(\partial T)$.
The proof is a simple blowup argument in which one assume the theorem is not true and gets the currents $$S^{(k)}=S_1^{(k)}+S_2^{(k)}\quad \|S^{(k)}\|=\|S_1^{(k)}\|+\|S_2^{(k)}\|,$$ violating the inequality $(*)$ with $\beta=1/k$. Then by compactness of minimizing currents, there is a subsequence so that $S^{(k)}\to S$ $S_i^{(k)}\to S_i$ $i=1,2$ with $\partial S=\partial S_1=\partial S_2=0$. The place where I get confused is that they directly conclude from this that $S=\partial E$ and $S_i=\partial E_i$ are oriented boundary of least area.
I can see that $S$ is the boundary of a Caccioppoli set $E$ since each $S^{(k)}$ is assumed to be so. But I cannot see why $S_i$ are both oriented boundary of some Caccioppoli set since they are a priori rectifiable currents. Even with $\partial S_1=\partial S_2=0$, I can only conclude from this that they are boundary of some $n+1$ dimensional rectifiable currents but it seems hard to show that they are actually some Caccioppoli set.
Maybe I missed some trivial points and I would like to thank you for help.