Little Hölder spaces - Reference request

The little Hölder space $c^\gamma(\mathbb{R}),$ for $0<\gamma<1,$ is usually defined in one of the following ways:

1. $c^\gamma(\mathbb{R})$ is the clousure of $C^\infty(\mathbb{R})$ in the usual Hölder norm of $C^\gamma(\mathbb{R})$.
2. $c^\gamma(\mathbb{R})$ is the closed subspace of $C^\gamma(\mathbb{R})$ consisting of those functions that satisfy the vanishing condition \begin{equation} \label{eq:vanishingCond} \lim_{h\to 0}\sup_{\substack{x,y\in\mathbb{R}\\ 0<|x-y|<h}} \frac{|f(x)-f(y)|}{|x-y|^\gamma}=0.\end{equation}

It is well known, almost folklore, that these two definitions are equivalent. This equivalence appears as an exercise in, for instance, "An Introduction to Harmonic Analysis" by Y. Katznelson.

Similar definitions exist for the little Zygmund class as well.

I'm looking for a reference to cite that proves these equivalences. Any ideas?

Thanks in advanced!