Show that no injection exists for those infinite sets

[andytoh]

Let A,B,C be infinite sets. Suppose there is no injection from A to B and no injection from B to C. Prove there is no injection from A to C (without using cardinality and Schroeder-Bernstein).

My current solution:
Let f:A-> C. Assume f= s.r (. means composition), where r:A-> B and s:B-> C. If f is injective, then so is r (already proven), a contradiction. But what if we can't assume f= s.r?

 

[andytoh]

So f=s.r must be true?

 

[andytoh]

So, I'll stick with my current proof.