Look, this is not a trivial problem. It usually makes use of the pigeon-hole principle.
You are not going to find a trivial proof. Any proof is equivalent to proving a form of the pigeon-hole principle.
That said, try this.
Suppose that is injective.
is collection of pair-wise disjoint subsets of .
But by the nature if functions .
From that how can conclude that is surjective?