I was given feedback for a homework problem,
Let $G$ be a group. Let $H$ be a subgroup of $G$. Let $a \in G$. Let $Ha$ be a subgroup of $G$.
The problem was to show that $Ha = H$.
I established that $H \subset Ha$ and then I made the argument that since $Ha = \{ha \vert h \in H\}$, $\vert Ha \vert \leq \vert H \vert$ so $H = Ha$
I was marked wrong and given the justification that $\vert Ha \vert \leq \vert H \vert$ isn't valid if $\vert H \vert = \infty$
I wanted to understand why, as I think the point still holds that $Ha$ can have at most as many elements as $H$, so if $H$ is a subset of $Ha$ then $H = Ha$.