About the Cantor-Schroeder-Bernstein theorem. If $A$ and $B$ are finite sets it is true that $|A|\leq |B|$ iff there is an injection from $A$ to $B$ and $|A|\geq |B|$ iff there is an surjection from $A$ to $B$. This motivates me to define these inequalities for arbitrary sets.
But doesn't this definition makes the Cantor-Schroeder-Bernstein theorem trivial?
What is wrong here? I should have to prove the Cantor-Schroeder-Bernstein theorem before defining these inequalities for arbitrary sets? Or they don't make sense at all?
 A: First, let me point out that if $B$ is empty and $A$ is not empty, then $|A|\geq|B|$, but there are no surjections from $A$ onto $B$.
Secondly, for infinite sets, it's entirely unclear why $|A|\leq|B|$ and $|B|\leq|A|$ should imply $|A|=|B|$. It's easily enough when you can actually write down the functions by hand. But if I had just told that there is an injective function from $\Bbb Q$ into $\Bbb N$, how would you propose to turn it into a bijection "off the cuff"?
Sure, in the case of $\Bbb N$ it's relatively easy. So let's press on. How would you suggest to construct a bijection between $\Bbb Q$ and the algebraic numbers? Or between $\ell_5$ and $\ell_2$?
As the comments point out, the main difficulty is to prove the theorem without using the axiom of choice. Cantor somewhat dismissed it as trivial once the axiom of choice is given, or rather the well-ordering theorem. And indeed this is almost trivial if you know that every set has a minimal ordinal with which it can be bijected and the basic properties of ordinal comparability.
But without the axiom of choice? Without the axiom of choice things become much more difficult. And in fact, the statement "$|A|\geq|B|$ if there is a surjection from $A$ onto $B$ [or $B$ is empty]" is no longer true, because there might not be an injection from $B$ into $A$, despite there is a surjection from $A$ onto $B$.
The triviality is somewhat of a cultural consequence. You've been born—mathematically speaking—into a world where the theorem is well-established; possibly you're a second, third or even fourth generation of people who learned mathematics when the theorem was already an established result of mathematics. This makes you far more susceptible to taking it for granted. Just as a kid today if they can imagine a world without televisions, or if they are young enough, a world without smartphones, the internet, Google. For them it's as trivial as the Cantor–Bernstein theorem is to you.
A: If you define the relation $\le$ on arbitrary sets by 
$$ |A|\le |B| :\iff \text{ there is an injection }A \to B $$
then you need the Cantor-Schröder-Bernstein theorem to prove (that is something you have to prove) that $\le$ is an order relation. Just because you call something $\le$, it does not mean that 
$$ |A|\le |B|, \ |B|\le |A| \implies |A|= |B| $$
holds true. That is Cantor-Schröder-Bernstein.
