Cardinality: Injection between subsets of Uncountable set

Question

assuming, S is infinite uncountable, I am trying to come up with injective f: (S union N) -> S. Where N is naturals. So far I created S0 which consists of infinite sequence of elements of S, such that S0 = {s1,s2,s3, .... } that way I can have injective f1: N -> S0. But I am having hard time trying to prove there exist injective f2: S -> S/S0.

Answer 1

Assume N and S are disjoint.
For each s in { s1, s2, s3,.. } pull from S a sequence { t$_s$1, t$_s$2, t$_s$3,.. } with t$_s$1 = s. Do this in a manner that none of the sequences have a common element.
This can be done because S is uncountable and all those sequences are requiring a mere countable number of elements.

To create a bijection from S - S0 to S, for each s in { s1, s2, s3,.. }
map t$_s$2 to t$_s$1, t$_s$3 to t$_s$2, etc.
and for all of those elements not in any of the sequences, map them to themselves.

The case that S and N are not disjoint is left for the diligent reader.