I need to prove that continuously differentiable functions on $\,\left[a,b\right]\,$ are a subset of Hölder space of order $1$ and that the Hölder space of order $1$ is a subset of the Hölder space of order $\mu$, $\;0 < \mu < 1,\,$ i.e. $\,C^1 \subset H_1 \subset H_\mu$.
Can anyone point me in the direction of a book or a theorem that proves this?