I came across the following problems on limit supremums and infimums:
Let $(A_n)$ be a sequence of subsets of $X$. Define $$\text{lim sup} \ A_n = \{x \in X: x \in A_n \ \text{frequently} \}$$ and $$\text{lim inf} \ A_n = \{x \in X: x \in A_n \ \text{ultimately} \}$$
Show that $$\text{lim inf} \ A_n \subset \text{lim sup} \ A_n$$
$$\text{lim inf} \ A_n = \bigcup_{n=1}^{\infty} \bigcap_{k=n}^{\infty} A_k$$
$$(\text{lim sup} \ A_n)' = \text{lim inf} \ A_{n}'$$ $$\text{lim sup} \ A_n = \bigcap_{n=1}^{\infty} \bigcup_{k=n}^{\infty} A_k$$
Note that $x_n \in A_n$ frequently means that $(\forall N) \ \exists n \geq N \ni x_n \in A_n$. Also $x_n \in A_n$ ultimately means that $\exists N \ni n \geq N \Rightarrow x_n \in A$.
The first follows since for $x \in \text{lim inf} \ A_n$ then $x \in A_n$ ultimately which means that it is contained in $\text{lim sup} \ A_n$. For the last three, would I just use DeMorgan's laws and the definition of unions and intersections to deduce that they are the same as the definitions given above?