From the wikipedia page on Holder spaces it says that if $0 < \alpha < \beta \leq 1$, then there is an inclusion map $\iota : C^{0 , \beta, }(\Omega) \rightarrow C^{0 , \alpha}(\Omega)$, where $\Omega$ is a bounded subset of $\mathbb{R}$.
My question is how to show this explicity. Say, take $f \in C^{0 , \beta, }(\Omega)$. Then there is $K > 0$ such that, for all $x,y \in \Omega$, $| f(x) - f(y) | \leq K | x - y |^\beta$. If we can get some constant $C > $ such that $ K | x - y |^\beta \leq C | x - y |^\alpha, \forall x,y \in \Omega$ we'd be done because then we would have $| f(x) - f(y) | \leq C | x - y |^\alpha$. Could we take $C > K \cdot \sup_{x,y \in \Omega}\frac{|x - y|^\beta}{| x - y |^\alpha}$? How do we know this is definitely finite.
Thanks for any help, I think I'm really missing something simple.