Alexandre has answered correctly, but I wanted to add that the norm closure of $C^\infty[0,1]$ in the $\alpha$-Holder space is the "little" $\alpha$-Holder space (a.k.a. little Lipschitz space) consisting of those $\alpha$-Holder functions that are locally flat in the sense that $\lim_{y \to x}\frac{|f(x) - f(y)|}{|x - y|^\alpha} = 0$ for all $x \in [0,1]$. It's easy enough to see that every $C^\infty$ function satisfies this condition and that the set of locally flat $\alpha$-Holder functions is norm closed. (Alexandre's function $|x|^\alpha$ fails to be locally flat at $x = 0$.) To get density we need to use "uniform separation of points"; see Corollary 8.30 of my book {\it Lipschitz Algebras} (second edition).

Alexandre has answered correctly, but I wanted to add that the norm closure of $C^\infty[0,1]$ in the $\alpha$-Hölder space is the "little" $\alpha$-Hölder space (a.k.a. little Lipschitz space) consisting of those $\alpha$-Hölder functions that are locally flat in the sense that $\lim_{y \to x}\frac{|f(x) - f(y)|}{|x - y|^\alpha} = 0$ for all $x \in [0,1]$. It's easy enough to see that every $C^\infty$ function satisfies this condition and that the set of locally flat $\alpha$-Holder functions is norm closed. (Alexandre's function $|x|^\alpha$ fails to be locally flat at $x = 0$.) To get density we need to use "uniform separation of points"; see Corollary 8.30 of my book Lipschitz Algebras (second edition).