I was solving a question where I had to state whether the subset of Q in interval [1,3] (Let's call it set B) is finite or not, and it dawned to me that since Q is countable and infinite, it has a bijection with N.
If we take every element of N that correspondence to every element of set B and create a 1-1 correspondence between that subset and another subset of N (Let's call it set A) in the form {1,2,3,...,n} for some n natural number, we can in turn create a bijection between set A and set B.
Meaning that B can be written in the form B = {a1,a2,a3,...,an} such that all elements are listed and n is some natural number. Which is the definition of a non-empty countably finite set in my course.
Is this true? And is the above a reasonable proof? Would this work for interval (1,3) as well?
Counter-example: $$\Bigl\{\,\frac1n\:\Bigm\vert\: n\in\mathbf N^*\,\Bigr\}.$$