Prove that $f(\limsup A_n)\subset \limsup f(A_n)$. Give an example where $f(\limsup A_n)\neq \limsup f(A_n)$.
I guess where I'm having trouble with this is what exactly does it mean to $f(\limsup A_n)$. I know what the $\limsup A_n$ but when it turns into a function then I dont know where to go from there.
$f(\lim \sup A_n)=\{f(x): x\in \lim \sup A_n\}.$
We have $$y\in f(\lim \sup A_n)\implies$$ $$\implies \exists x \in \lim \sup A_n \;(y=f(x))\implies$$ $$\implies \exists x (y=f(x)\land \{n:x\in A_n\} \text { is infinite })\implies$$ $$\implies \exists x \;(\{n:y=f(x)\in f(A_n)\} \text { is infinite }) \implies$$ $$\implies \{n: y\in f(A_n)\} \text { is infinite})\implies $$ $$\implies y\in \lim \sup f(A_n).$$