In our PDE seminar, we met the same kinds of questions, and we think the answer is "WRONG". The smooth functions is NOT dense in H{"o}lderHölder spaces.
An example is, $$f(x) = |x|^{1/2} \quad x \in (-1,1)$$ it is easy to check that $f$ is $1/2$-H{"o}lderHölder continuous.
For details,
for any $g \in C^{1}((-1,1))$, then the derivative of $g$ is continuous
at $0$, so we have
$$
\lim_{x \to 0} \frac{|g(x)-g(0)|}{|x|^{1/2}} = \lim_{x \to 0}
|x|^{1/2}\frac{|g(x)-g(0)|}{|x|} = 0
$$
and
$$
\omega_{1/2}(g-f) \ge \frac{|(g(x)-f(x))-(g(0)-f(0))|}{|x|^{1/2}} \ge
|\frac{|(g(x)-g(0)|}{|x|^{1/2}}-\frac{|f(x)-f(0)|}{|x|^{1/2}}|
$$
but
$$
\frac{|f(x)-f(0)|}{|x|^{1/2}}=1 \quad x \in (-1,1) \quad x \neq 0
$$
let $x \to 0$, we obtain $\omega_{1/2}(g-f) \ge 1$.
Thus, for any $g \in C^{1}((-1,1))$, we have $\omega_{1/2}(g-f)\ge 1$.
For $0< \alpha <1$ we can make similar examples, but when $\alpha = 1$, the proof of the counter-example may be different.
