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 \lim_{n\rightarrow \inf} s_n = s , then \lim_{n\rightarrow \inf} cs_n = cs.
(b) Prove that if \lim_{n\rightarrow \inf} s_n = s then \lim_{n\rightarrow \inf} s^2_n = s^2. (Hint:recall that convergent sequences are bounded.)
(c) Use the polarization identity ab = \frac{1}{4}((a+b)^2-(a-b)^2) to prove that if \lim_{n\rightarrow \inf} s_n = s and \lim_{n\rightarrow \inf} t_n = t , then \lim_{n\rightarrow \inf} t_ns_n = ts.


my work:
(a) Since for all \varepsilon > 0 , there exists an N, such that if n > N , |s_n-s| < \varepsilon , there should
exist an n_{\mu} \geq n such that c|s_n-s| \leq \varepsilon hence \lim_{n\rightarrow \inf} cs_n = cs

(b) \lim_{n \rightarrow \inf}s_n = s implies that {s_n} is compact and bounded. Take \varepsilon > 0. For some M, if n \geq M, then |s_n-s|=|s-s_n| < \sqrt{\varepsilon} and therefore |s-s_n|^2 < \varepsilon. so now \lim_{n \rightarrow \inf}(s_n-s)^2 = 0.
Take |s_n^2-s^2| = |s_n(s_n-s)-s(s-s_n)| \leq |s_n||s_n-s|-|s||s-s_n| \leq |s_n-s|(|s_n|-|s|) \leq \varepsilon which gives us \lim_{n \rightarrow \inf}(s_n^2-s^2)=0

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