Show that $H = \bigcup_{n > 0} S_n$ is not equal to Sym$(X)$ for $X = \{1,2,3,4,...\}$. Let $X = \{1,2,3...\}$ be the set of positive natural numbers, $S_n$ the permutation group, and Sym$(X)$ the set of all bijections from X to X with operation composition. I have the following questions.
1) How do I show that $H = \bigcup_{n > 0} S_n$ is a subgroup of Sym$(X)$, or how do I interpret this being a subgroup (in terms of compositions of bijections??).
2) How do I show that $H$ and Sym$(X)$ are not equal?
EDIT:
1) I have problems with the subgroup properties. 
First property.
The identity element of $S_2$ is not equal to the identity element of $S_4$ for example. I see that both are bijective functions such that $f(x) = x$. But $S_2$ sends $(12) \to (12)$, but does nothing with $(1234)$, it's not in its domain? So how do I prove that there exists a $id_f \in H$, such that $id_f$ in all $S_n$, and that this one equals the one in $Sym(X)$. 
Second property.
If $a,b \in \bigcup_{n > 0} S_n$, then $a,b \in S_i$ for some $i \leq n$, and thus $ab \in S_i$, and thus $ab \in \bigcup_{n > 0} S_n$. But what if $a$ and $b$ are not in the same $S_i$? I can only assume that the union of subgroups is a subgroup if $S_1 \subset S_2 \subset ... \subset S_n$.
Third property.
This one is easy.
2) At first I was confused because I thought that you can always find enough bijective functions in $H$. But each $S_n \in H$ has finite order, and thus works on finitely many elements in $\mathbb{N}$. Thus a function $f \in Sym(X)$ that works on all $x \in \mathbb{N}$ cannot be in $H$, even though (I believe) all $f \in Sym(X)$ have finite order too... I think I understand this part now.

Hint: Note that $H$ is the set of permutations of $X$ that have finite support in the sense that they move just a finite subset of $X$ and fix the rest.
The hint proves both points:


*

*If $\sigma, \tau$ have finite support than so do $\sigma^{-1}$ and $\sigma\tau$. The identity clearly has finite support.

*Not all permutations of $X$ have finite support. For instance, the one that swaps even and odd numbers does not.
