Weak * Convergence and Stone-Weierstrass Theorem I'm trying to solve the first part of the following exercise:

Sorry for being lazy, but I am new to using stackexchange and still learning  to type LaTeX. 
I am trying to prove the first part using epsilon-partition language, but got stuck finding the actual partition and the desired N.
To be honest this is a homework problem, so please just leave some hints.
Thanks!  
 A: Here are some hints.
First, I'm pretty sure there is a typo in the definition of $c_n$. It should read:
$$c_n = \left(\int_{-1}^1 (1-x^2)^n dx\right)^{-1}.$$
Note that (from the definitions) $d\nu_n=c_n(1-x^2)^n dx$ and $d\mu$ is the Dirac measure at $0$. 
Therefore what we have to show to prove weak-* convergence is
$$\lim_{n\rightarrow\infty} \int_{-1}^1 f(x) P_n(x) dx = f(0),\hspace{2cm}(*)$$
where $P_n(x)=c_n(1-x^2)^n$ and $f$ continuous on $[-1,1]$.
To see how to do that try to draw how $P_n$ looks like as $n$ grows. Also note that $\int_{-1}^1 P_n(x) dx = 1$. Additionally, you will need to use uniform continuity of $f$. This should help you to get an idea how to prove the claim (if not, I can provide more hints).
Now a final hint regarding the proof of Weierstrass' approximation theorem from this: fix $y\in [-1,1]$. The above procedure approximates $f(0)$. Can you change it to approximate $f(y)$ instead? Then it remains to notice that the approximating numbers vary polynomially in $y$ and argue why uniform convergence holds.
Edit: Here is how to prove (*):
Write $f(0) = \int_{-1}^1 f(0) P_n(x) dx$. For $\delta>0$ we split
$$\int_{-1}^1 (f(x)-f(0)) P_n(x) dx = \int_{|x|\ge\delta} (f(x)-f(0)) P_n(x) dx + \int_{|x|<\delta}(f(x)-f(0)) P_n(x) dx$$
Treat the first summand using that $(1-x^2)^n$ tends to $0$ if $x$ is away from $0$ and $n\to\infty$. For the second summand use uniform continuity of $f$ (I'll leave you the details).
