I would like to prove myself that:
$$\overline{C^{\infty}}^{||\cdot||_{\alpha}} \neq C^{\alpha}$$
where the left hand side is the closure with respect to the semi-norm of Holder continuous functions (see below). The right hand side being $\alpha$-Holder continuous functions, with classical semi-norm:
$$ ||X||_{\alpha} = \sup \frac{ X_t - X_s } {(t-s)^{\alpha}} $$
During the lecture, the professor mentioned that $X_t = t^{\alpha} $ is a counter example.
I have proven that $t^{\alpha}$ is in $\alpha$-Holder continuous functions.
Now I struggle to prove that the characterisation of the LHS is not respected by $t^{\alpha}$:
$$\lim_{ \delta \to 0} \sup_{ |t-s| < \delta } \frac{ X_t - X_s } {(t-s)^{\alpha}} = 0 $$
My attempt is to pick $t = s + \delta/2$. Now I would need help to find a lower-bound for:
$$ \frac{|(s-\delta/2)^{\alpha} - s^{\alpha}|}{(\delta/2)^{\alpha}}$$