I've just started measure theory and I'm really struggling with what's supposed to be easy proofs. Any help would be appreciated.

1.1) Let $\displaystyle (E_n)_{n\epsilon N}$ be a sequence of subsets of X. Prove that $\displaystyle \limsup E_n =\bigcap _{k=1}\bigcup_{n\geq k} E_n$

1.2) Let $\displaystyle (E_n)_{n\epsilon N}$ be a sequence of subsets of X. Prove that $\displaystyle \liminf E_n =\bigcup _{k=1}\bigcap_{n\geq k} E_n$

I attempted to do this using the definition of limsup and liminf (for limsup. x belongs to limsup if it belongs to infinitely many E_n and y belongs to liminf if it belongs to all but finitely many E_n) but my professor said to rather do it using set inclusion that is if x belongs to limsupE_n then x belongs to $\displaystyle \bigcap _{k=1}\bigcup_{n\geq k} E_n$ and if y belongs to $\displaystyle \bigcap _{k=1}\bigcup_{n\geq k} E_n$ then y belongs to limsup E_n. I started by assuming $\displaystyle (x\epsilon limsup E_n$ and $\displaystyle y\epsilon \bigcap _{k=1}\bigcup_{n\geq k} E_n$ where do I go from here?