Injection - Bijection function Question Suppose $A, B$ are two non-empty sets and let $$f:A\to B$$ and $$g:B\to A$$ be two injective (one to one) functions.
QUESTIONS:
1. Is there exist a bijective function from $A$ to $B$?
2. If so, how can we prove it?  
I think, there should be a such function. But I can not come up with an argument to prove it?

I have seen 2 proofs, one complicated, and this one: For $x\in A$ the possible sequence $$x,\; f^{-1}(x),\; g^{-1}f^{-1}(x),\; f^{-1}g^{-1}f^{-1}(x),...$$ may have just one term (if $f^{-1}(x)$ doesn't exist), or 2 terms (if $f^{-1}(x)$ exists but $g^{-1}f^{-1}(x)$ does not), et cetera. 
Let $x\in E$ if the sequence stops after an even number of terms. Let $x\in O$ if the sequence stops after an odd number of terms. Let $x\in I$ if the sequence has no end.
Let $h(x)=f(x)$ if $x\in O.$ Let $h(x)=f^{-1}(x)$ if $x\in E\cup I.$ I will leave it to you to verify that $h:A\to B$ is a bijection.
I have seen this referred to in books as the Schroeder-Berstein Theorem, the Cantor-Bernstein theorem, and the C-S-B theorem.
