Let :

1) Suppose f is 1-1 ==> the set has as many elements as A (and as B) ==> , otherwise B is a finite set having a proper subset with its same cardinality and this is the definition of INFINITE sets ==> f is onto.

2) Suppose now that f is onto ==> ; if then , which is impossible since , thus it must be that ==> f is 1-1.

Tonio