I have this exercise and I don't know how to solve it Let $(A_n)_n \subset (E,d),(B_n)\subset (F,d')$ 2 sequence of sets from a metric space,and $f: E \rightarrow F$ a continuous aplication .
Prove that :
1)$\displaystyle f(\limsup_{n\to\infty} A_n) \subset \limsup_{n\to\infty} f(A_n)$
2)$\displaystyle f(\liminf_{n\to\infty} A_n) \subset \liminf_{n\to\infty} f(A_n)$
3)$\displaystyle \limsup_{n\to\infty}f^{-1}(B_n) \subset f^{-1}(\limsup_{n\to\infty} B_n)$
4)$\displaystyle \liminf_{n\to\infty}f^{-1}(B_n) \subset f^{-1}(\liminf_{n\to\infty} B_n)$
with: $x\in \overline\lim(A_n) \Rightarrow$ $ x\in \displaystyle\bigcap_{\varepsilon>0}\bigcap_{N>0}\bigcup_{n\geq N} (A_n)_\varepsilon \Rightarrow \forall \varepsilon >0, \forall N>0, \exists n\geq N ;d(x,A_n)< \varepsilon$
help me please , thank you .
Hint: See, Own Lecture Notes Functional Analysis. Theorem 2.15.1, p17.