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.

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).
