Prove the set A = {1, 1/2, 1/3, 1/4,...} is infinite.
I did by showing A is equivalent to N, the set of natural numbers. So we have to show there exists a bijective function between A and N. Lets take:
F: N -> A => F(x) = 1/x
Clearly, this is one to one since: F(x) = F(y) => 1/x = 1/y => x = y
I'm having trouble wrapping my head around the onto part.
We can make y the subject to get 1/x and this gives: F(1/x) = 1/(1/x) = x but the thing is 1/x is NOT in N. So how clearly something isn't right.
This isn't the only case either as I've often run into trouble proving onto as after making x the subject the result doesn't seem to belong to the domain (i.e. N).
How do I correct this and what would be the proper way to do this?
You are on the right track, you have already proved the function $F$ is injective. Let's take any element $a$ from $A$, by definition there exists a positive integer $N$ such that $N=\dfrac1a$, thus $a=\dfrac1N=F(N)$, we can conclude the function $F$ is onto.