Suppose $\bigcup_{n=1}^{\infty}A_n $ has cardinality continuum, prove that at least one $A_n$ has cardinality continuum.
If the choice axiom (C.A.) holds, König's theorem can be used to prove it; if the continuum hypothesis (C.H.) holds, then proof by contradiction can be used; if neither the choice axiom nor the continuum hypothesis is used, can the proposition still be proved?