Hey guys I'm trying to prove the following:
$X=\emptyset \lor$ There is a surjection $g: \mathbb{N} \rightarrow X \implies X$ is finite $\lor$ there is a Bijection from $\mathbb{N}$ to X
I did case distinction where I assumed that $X$ is finite where the implication is obviously true. In the case $X$ is infinite I want to show that there is a injection from $N$ to $X$ but also a surjection from $N$ to X and via Schrödinger-Bernstein we have a Bijection. But for this I need the implication $X$ is infinite thus there is an injection from $\mathbb{N}$ to X. There are already questions answered similar to this one but not precisely asking what I'm asking. Can anyone help?
We will define a function $f: \mathbb{N} \to X$
Let $x_0$ be an element of $X$. We define $f(0)=x_0$. Since $X$ is infinite, then $X-\{x_0\} = X_1$ is also infinite. We pick $x_1$ from $X_1$ and let $f(1) = x_1$.
By iterating this process we get an injection from $\mathbb{N}$ into $X$