Let $A$ be a nonempty subset of $\Bbb N$ without a greatest element. Then there exists a unique, strictly increasing, and surjective mapping $f:\Bbb N \to A$. In my textbook, the author said that this theorem is very important since many other theorems regarding the countability depend on it...

Let $A$ be an infinite subset of $\Bbb N$. Then there exists a bijection from $A$ to $\Bbb N$. My attempt: We define $f:A \to \Bbb N$ by $f(a)=|\{a'\in A\mid a'<a\}|$. $f$ is injective For $a_1,a_2\in A$ and $a_1<a_2$, then $\{a'\in A\mid a'<a_1\} \subsetneq \{a'\in A\mid a'<a_2\}$, t...