Polynomials not dense in holder spaces How to prove that the polynomials are not dense in Holder space with exponent, say, $\frac{1}{2}$?

By exhibiting a function that cannot be approximated by polynomials in the norm of $C^{1/2}$, such as $f(x)=\sqrt{x}$ on the interval $[0,1]$. The proof is divided into steps below; you might not need to read all of them.
Let $p$ be a polynomial.  

 $(p(x)-p(0))/x\to p'(0)$ as $x\to 0^+$



 $|p(x)-p(0)|/x^{1/2}\to 0$ as $x\to 0^+$ 



 $|(f(x)-p(x))-(f(0)-p(0))|/x^{1/2}\to 1$ as $x\to 0^+$ 



 $\|f-p\|_{C^{1/2}}\ge 1$

