A group containing a copy of $S_n$ for $n = 1,2,\cdots$ Suppose that $G$ is a group containing a copy of $S_n$ for each $n$. Does it mean that $G$ contains a copy of $S_\Bbb{N}$?

Not necessarily, consider the group $G$ of finitary permutations of $\mathbb{N}$, that is, those elements of $S_{\mathbb{N}}$ that fix all but a finite number of elements, that is,
$$
G = \{ g \in S_{\mathbb{N}} : \text{the set $\{ a \in \mathbb{N} : g(a) \ne a \}$ is finite}\}.
$$
