Let $H \subset C[0,1]$ be the set of all Holder-continuous functions under the uniform norm $||\cdot||_\infty$, i.e. $$H = \bigcup_{\alpha > 0}C^{0, \alpha}[0,1]. $$ Is $H$ closed in the uniform norm?
I don't believe that this set is closed, as if $f_n \to f$ in $||\cdot||_\infty$ norm and $f_n$ is Holder-continuous with $$|f_n(x) - f_n(y)| \leq C_n \cdot |x-y|^{\alpha_n}, \forall x,y \in [0,1], $$ then I don't think we must have that $C_n$ or $\alpha_n$ converge. However, I have not been able to come up with a counter example.
Every polynomial is Holder continuous since it has a bounded derivative. Every continuous function is the uniform limit of a sequence of polynomials. So take any continuous function which is not Holder continuous and you will see why the set is not closed.