please help with limit proof

I have a 3 parter. I think I have the first part, the second part I am not sure of and the third part I am kind of getting desperate with...

(a) Prove first that if c is a constant, and , then .

(b) Prove that if then . (Hint:recall that convergent sequences are bounded.)

(c) Use the polarization identity to prove that if and , then .

my work:

(a) Since for all , there exists an , such that if , there should

exist an such that hence

(b) implies that is compact and bounded. Take . For some , if , then and therefore . so now .

Take which gives us

(c) If my other two are even correct, this is where I am stuck. PLEASE give me a little nudge...