As the title says, that's all my question. Let me state it again:
Is it true that every infinite set has an infinite countable subset?
It seems so trivial, my thought goes like this: pick an arbitrary element and denote it as $x_1$; pick the next one and denote it $x_2$, and so on.
Is my proof correct? Since it seems so simple, I'm not sure of it.
To avoid any further confusion, the definitions used are:
Finite: In bijection with $\{1\ldots n\}$ for some $n$.
Infinite: Not finite.
Countably infinite: In bijection with $\Bbb N$.
Countable: Finite or countably infinite.
Best Answer
You are on the right track, however you cannot use the phrase "the next one". The idea is:Let $S$ be an infinte set. Pick an element $x_1\in S$, then since $\{x_1\}$ is finite, $S\setminus\{x_1\}\ne\emptyset$, so pick an element $x_2\in S\setminus\{x_1\}$ and so on.
Note the Axiom of Choice is involved here.