Show that every infinite set is at least as large as N. Which implies Card(A) >= Card(N) for every infinite set A.

Question

I understand that A is bigger if there is an injective function f : N --> A. To show every infinite set is at least as big as N, how should I approach this question ? Thank you very much.

Answer 1

There are a couple of approaches, but the most straightforward is this: since $A$ is infinite it is nonempty, so choose any element and call it $x_1$. Then choose any element in $A\setminus \{x_1\}$ and call it $x_2$ and so on. This step needs to be justified - how do you know there are still elements in $A$? Once you've got that, it's a matter of continuing indefinitely to create a set $\{x_n\}_{n\in\mathbb{N}}$. This gives you the bijection you need.

The comments about choice are because you have to assume the axiom of choice in order to do this. But if you're just learning about cardinalities then it's best if you don't worry about why, for now.