• Sep 30th 2010, 04:19 PM
DontKnoMaff
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 $\displaystyle \lim_{n\rightarrow \inf} s_n = s$ , then $\displaystyle \lim_{n\rightarrow \inf} cs_n = cs$.
(b) Prove that if $\displaystyle \lim_{n\rightarrow \inf} s_n = s$ then $\displaystyle \lim_{n\rightarrow \inf} s^2_n = s^2$. (Hint:recall that convergent sequences are bounded.)
(c) Use the polarization identity $\displaystyle ab = \frac{1}{4}((a+b)^2-(a-b)^2)$ to prove that if $\displaystyle \lim_{n\rightarrow \inf} s_n = s$ and $\displaystyle \lim_{n\rightarrow \inf} t_n = t$ , then $\displaystyle \lim_{n\rightarrow \inf} t_ns_n = ts$.

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

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

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