Let $G$ be a group and suppose $E \subseteq G$ finite and symmetric (i.e. $g^{-1} \in E$, for all $g \in E$) . Define $E^n=\lbrace g_1g_2...g_n \mid g_i \in E \rbrace$, for all $n \in \mathbb{N}$. For all $g \in G$, denote $gE^n=\lbrace gt \mid t \in E^n \rbrace$. Is there an elementary way to show that $$|gE^n \Delta E^n| \leq |E^{n+1}| - |E^n|,$$ where '$\Delta$' denotes the symmetric difference?
$\textbf{EDIT}$:
I misread the the question. It should be $g \in E$.