in this case, i believe that you want to show this for an arbitrary ring (possibly non-commutative) R, *not* the ring of real numbers (because every real number centralizes every other real number, because multiplication is commutative on the reals).
you need to show 3 things:
1) if x,y are in C(a), then x - y is in C(a).
2) if x,y are in C(a) , so is xy.
3) C(a) is non-empty (so we can actually use (1) & (2)).
proving a is in C(a) will show (3). does aa = aa?
the distributive law should figure heavily in your proof of (1). the associative law of multiplication should figure heavily in your proof of (2).
to prove that:
is equivalent to saying that for r in C(R), ra = ar for EVERY a in R. is that not the definition of C(R)?