# Interesting bijection $\mathbb N\times\mathbb N\to\mathbb N$

$$(b)$$ 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.