Understanding points about countable sets from wikipedia

Question

I am struggling to understand following points made on wikipedia page of counting sets:

Let S be a set. The following statements are equivalent:

1. S is countable, i.e. there exists an injective function f : S → N.
2. Either S is empty or there exists a surjective function g : N → S.
3. Either S is finite or there exists a bijection h : N → S.

I understand how point 1 implies point 2. But I dont understand how point 2 implies point 3.

Let me prove $(1)\implies (2)$:

Let $f:S\rightarrow \mathbb N$ be injective function.
First of all $f$ is a function. So each element of $S$ is mapped to some element in $\mathbb N$, since any function maps all elements in its domain to its codomain. Since each element in domain has corresponding mapping element in codomain, if we make a new function $g$ with domain as codomain of $f$ and codomain as domain of $f$ , it will be surjective function, since by definition, in surjective function, all elements in codomain have corresponding mapping element in domain. Thus, $g:\mathbb N\rightarrow S$ is surjective function. (Is this proof correct?)

How can we prove $(2)\implies (3)$ ?

Comments

Check this, if it helps - https://gateoverflow.in/216802/set-theory

Answer 1

with 1+2=> prove 3

Comments

Well...I will honestly love if someone helps me prove this, pleaseeeee..... :)

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I have further doubt:

Let us consider following points:

(1) $f:S\rightarrow \mathbb N$ is injective function
(2) there exists surjective function $g:\mathbb N \rightarrow S$
(3) there exists bijective function $h:\mathbb N \rightarrow S$

If $(1)\implies (2)\implies (3)$, it means $(1)\implies (3)$. And since $h:\mathbb N \rightarrow S$ is bijective, its also injective. Does that means any injective function $f:S\rightarrow \mathbb N$, there also exists injective function from $\mathbb N$ to $S$, that is $\mathbb N \rightarrow S$?