# How to show that if there is a injective function $f : A \rightarrow B$, then there is a injective and monotone $g : A \rightarrow B$ too

We have $A,B \subseteq \mathbb{N}$ non-empty subsets of natural numbers. How to show that if there is a injective function $f : A \rightarrow B$, then there is a injective AND monotone $g : A \rightarrow B$ too.

And is it true that if there are injective functions $f : A \rightarrow B$ and $g: B \rightarrow A$ then there is also a bijective function $h: A \rightarrow B$

• At least as a first approach I would divide into cases according to whether $A$ is finite or infinite. (Later on it may be that you can combine both cases in one wording). – Henning Makholm Jan 5 '16 at 12:21
• For the second question: Yes; this is the celebrated Cantor-Schröder-Bernstein theorem. – Henning Makholm Jan 5 '16 at 12:22

The construction of the monotone injective function $g:A\rightarrow B$ is as follows: Let $A=\{a_1, a_2, ...\}.$ an ordered arrangement of $A$. Define $g(a_1)=\min f(A), g(a_2)=\min (f(A)- \{g(a_1\}),g(a_3)=\min (f(A)-\{g(a_1), g(a_2)\}) ...$
The answer to the second question is yes, it's the Schröder-Bernstein theorem. It's valid in general and do not require $A$ and $B$ to be subset of $\mathbb N$.