For a infinite set $S$, define $S^n=\{(s_1,...,s_n) | s_i \in S\ \forall 1\le i \le n \}$. Prove that there exists a bijection $S \to S^n$.
Here is my attempt:
By the Schroder-Bernstein theorem it suffices to find an injection $f:S \to S^n$ and an injection $g: S^n \to S$. Let $z^*$ be some fixed element of $S$.
The first part is easy: we can simply take $f(s)=(s,z^*,...,z^*)$ for every $s \in S$.
The second part can be proven by induction. Assuming the base case $n=2$ we can find an injection $h:S^2 \to S$. Now suppose we have found an injection $f:S^n \to S$. Define $f':S^{n+1} \to S$ by $f'(s_1,...,s_{n+1})=h(f(s_1,...,s_n),s_{n+1})$. Since $h$ and $f$ are injective, $f'$ is injective.
Thus all that remains is to prove the base case, i.e there exists an injection $S^2 \to S$ (equivalently there is a surjection $S \to S^2$). I'm not sure how to do this.
