# Prove that the set of natural numbers (in base 10) with exactly one of the digits equal to 7 is countably infinite.

Prove that the set of natural numbers (in base 10) with exactly one of the digits equal to 7 is countably infinite.

"In base 10" means that it's the natural numbers between 0 and 9, correct? What might the first step be in starting a formal proof?

I know that in order to prove cardinality, there must be a bijection, and that a set S is countably infinite if |S|=|$\mathbb{N}$|, but I'm not sure where those definitions would come into play in this instance.

• "I know that in order to prove cardinality, there must be a bijection" - true, but when all you need to show is 'countably infinite', it suffices to show 'not finite, and no bigger than $\mathbb N$' – AakashM Nov 16 '17 at 13:02

If $S$ is a subset of $T$, then $|S| \leq |T|$. The set in your question is a subset of the natural numbers, so it is either finite or countably infinite (and it's easy to see that it isn't finite).