Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Full question is in the title. It seems to me the answer is yes, because we can just order the numbers in the image of the function from starting from the least, and then there's 1-1 (is there?) correspondence with $\{1, 2, 3 , \dots \}$, but obviously this isn't a rigorous argument. Help is appreciated.

share|improve this question
add comment

1 Answer

Yes. If there is an injection $f\colon X\to\mathbb N$ we can define the following:

$$g(x) = |\{y\in X\mid f(y)<f(x)\}|$$

Where $|A|$ denotes the cardinality of $A$. This is a well-defined function because $g(x)$ is the cardinality of a bounded set of natural numbers, and therefore it is a natural number itself.

We can show by induction that $g$ is injective, and that its range is an initial segment of $\mathbb N$. If $X$ is infinite then $g$ is also surjective, because there is only one initial segment of $\mathbb N$ which is infinite... $\mathbb N$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.