$(b)$ Let $A$ be an infinite subset of the set of natural numbers $\mathbb N=\{0, 1, 2, ...\}$. Prove that there is a bijection $f\colon\mathbb N\to A$. <---- Have done this part with no bigger trouble

$(c)$ Let $A_n$ be the set of natural numbers whose decimal representation ends with exactly $n − 1$ zeros. For example, $71\in A_1, 70\in A_2$ and $15000\in A_4$. By applying the result of part $(b)$ with $A = A_n$, construct a bijection $g\colon\mathbb N\times\mathbb N\to\mathbb N$. <---- I cannot see an obvious of constructing required bijection


Define$$\begin{array}{rccc}B\colon&\Bbb N\times\Bbb N&\longrightarrow&\Bbb N\\&(m,n)&\mapsto&\text{$m$-th element of }A_n\end{array}$$Since each $A_n$ is an infinite subset of $\Bbb N$, and since $\bigcup_{n\in\Bbb N}A_n=\Bbb N$, this makes sense and $B$ is a bijection.

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  • $\begingroup$ That doesn't cover the "By applying the result of part (b)" part though. This hint seems missleading at best. $\endgroup$ – Olivier Roche May 7 at 10:39

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