Proving countability My professor tried to prove a claim that the given set is countable.
What confused me was that he first provided a surjection from $\mathbb{N}$ to the set and then added an additional step in proof to alter that surjection to transform into a bijection.
I thought providing a surjection from $\mathbb{N}$ to the set suffices to prove that the set is countable. 
Would there be a reason he would "transform" that surjection to provide a bijection instead?
 A: It depends on the theorems that you have proved so far, the definition that you have for countable, and of course a lot of unpredicted variables -- standing in front of a class, often you do things in a non-optimal way.
If you have defined a set to be countable if there is a bijection with $\Bbb N$, but not yet proved that it suffices to show that there is a surjection (and that the set is infinite), then the additional step is needed.
If you have shown that it suffices to have a surjection, then perhaps this was caused by the last reason I mentioned.
A: Perhaps the point was to show that the set was countably infinite; that would need the bijection. Or maybe he just wanted to emphasize that if the set is infinite and a surjection exists, then there will be a bijection, too. ANyhow, you're right that the surjection is sufficient.
A: If you define a countable set to be a set $S$ for which you can find a bijection between $S$ and a subset of $\mathbb N$ then you definitely meet to prove a bijection rather than a surjection.
There are foundational issues involved here, which go a little deeper than immediately obvious. You might want to think why the Cantor-Shroeder-Bernstein Theorem needs to be proved rather than being completely obvious. This is expressed in terms of injections rather than surjections - but we expect these to be dual concepts.
I think your professor may be encouraging you to work to the precise definition, rather than the apparently obvious corollary. Some of these things are more slippery than we first imagine. Taking care is a good habit.
