I need to prove that the set of finite decimals is countable. I know a set is countable if its cardinality is the same as that of the natural numbers which also means that there exists a bijective map between the set and the natural numbers. So as long as every element in the set is counted once and i have s systematic way of listing each member of the set it should prove that the set is countable, correct?

Well, i am not sure if this "counts" as a systematic method of listing the finite decimals and thus proving that they are countable. So what i thought of was that since i know already that the set of integers is countable, i can list the set of integers + 1 decimal point, then the set of integers + 2 decimal points, and so on until i have listed every finite decimal. is this a valid "systematic way" of listing every finite decimal?