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I'm studying Real Analysis using Robert Bartle's text Elements of Real Analysis.

Wonder if I could get some help on understanding the proof of the following theorem:

The following statements are equivalent: (a) S is a countable set. (b) There exists a surjection of $\Bbb{N}$ onto S. (c) There exists an injection of S into $\Bbb{N}$

Proof. (a) $\Rightarrow$ (b) If $S$ is finite, there exists a bijection $h$ of some set $\Bbb{N}_n$ on $S$ and we define H on $\Bbb{N}$ by

$$ H(k) = \left\{ \begin{array}{ll} h(k) & \quad k = 1,...,n, \\ h(k) & \quad k > n \end{array} \right. $$

Then $H$ is a surjection of $\Bbb{N}$ onto $S$.

If $S$ is a denumerable, there exists a bijection $H$ of $\Bbb{N}$ onto $S$, which is also a surjection of $\Bbb{N}$ onto $S$.

Question:

The part of the proof that I do not understand is: $h(k)$ for $k>n$. Why is there a need for this piecewise of the function? How is it defined? In the sense that, if set $S = \text{{A,B,C,D}}$, then there exists a bijection $h$ of set $\Bbb{N}_4$ on $S$ (which is then, $H$ is a surjection of $\Bbb{N}$ onto $S$). Why then $h(k)$ for $k>n$? How to map a surjection of $\Bbb{N}$ onto $S$ for $k>4$ when $S$ only have 4 elements?

I feel like I'm missing a point here but couldn't quite point out what it is?

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As you correctly observe, it does not make sense to define $H(k)=h(k)$ for $k>n$ since $h(k)$ is not defined in that case. I'm guessing this is just a typo, and it means to say $h(1)$ instead of $h(k)$ in that case. That is, you pick some particular element of $S$ (say the first one according to $h$, namely $h(1)$) and define $H(k)$ to be that particular element for all $k>n$.

(Incidentally, there is still an error here: $h(1)$ is only defined if $n>0$. Indeed, the statement being proved is not quite true: you need to assume $S$ is nonempty. If $S$ is empty, then $S$ is countable, but there is no surjection $\mathbb{N}\to S$, since there is no function $\mathbb{N}\to S$ at all.)

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