I'm currently working on this paper by Helfgott for a small project: https://arxiv.org/abs/1303.0239. After Lemma 3.1 (Ruzsa inequality) he says (and shows partially) that for any finite subset $A$ of any group $G$, we have $$[3.2]\;\;\;\;\;\; \frac{|(A\cup A^{-1}\cup\{e\})^3|}{|A|}\leq \left(3\frac{|A^3|}{|A|}\right)^3.$$ Furthermore he also says that adding the hypothesis $A=A^{-1}$, we can get $$ [3.3]\;\;\;\;\;\;\frac{|A^k|}{|A|}\leq\left(\frac{|A^3|}{|A|}\right)^{k-2}.$$
Can anyone help me completing the proof of [3.2] (and eventually [3.3]) in some simple way? The first part of the proof can be seen on page 12 of the paper. Thanks a lot.