How to show that L*=(L*)*?
It's a big problem if you can't tell whether your proof counts as a proof or not. Perhaps it's better to write the proof "in words". Here is an example.

Since $M \subseteq M^*$ for every language $M$, clearly $L^* \subseteq (L^*)^*$. Now suppose $w \in (L^*)^*$. Then $w = w_1 \ldots w_\ell$ for some $w_1,\ldots,w_\ell \in L^*$. Then for each $i$, $w_i = w_{i,1} \ldots w_{i,\ell_i}$, where $w_{i,j} \in L$. So $w = w_{1,1} \ldots w_{1,\ell_1} \ldots w_{\ell,1} \ldots w_{\ell,\ell_\ell} \in L^*$. This shows that $(L^*)^* \subseteq L^*$, and so $L^* = (L^*)^*$.

This is probably very similar to the proof you give, but your proof is written "in symbols" and so it is somewhat hard to parse.