Note that a set $A$ is said to be extremely infinite iff for each set $a$ : $|A\cup\{a\}| = |A|$.
Let $\alpha$ be a set.
Prove that if $\alpha$ is extremely infinite, then there exists some $\beta \subset \alpha$ (in particular $\alpha \neq \beta$) s.t $|\alpha| \leq |\beta|$
Since we should define such an injective and we only know that one property about $\alpha$, I assume that I just don't see it. I tried to use a diagram to simplify the problem however it just made it more vague.
Wish to get some orientation, thanks.
Take some $a$ with $a\notin\alpha$, e.g. $\alpha$ itself. Then you get a bijection $f:\alpha\cup\{a\}\to\alpha$ by $\alpha$ being extremely infinite. Set $\beta:=f[\alpha]$, which is a proper subset since it doesn't contain $f(a)$. By restricting $f$ to $\alpha$, you get a bijection between $\alpha$ and $\beta$, hence $|\alpha|\le|\beta|$.