How can we show that the space $C^{0, \gamma}(U)$ is complete??
I have tried the following:
So that the space is complete, the following has to stand:
$$\forall \epsilon >0, \exists n_0 \text{ such that } m, n \geq n_0 : ||u_m-u_n||_{C^{0, \gamma}} <\epsilon$$
We have that $$||u||_{C^{0, \gamma}}=||u||_{C(U)}+[u]_{C^{0, \gamma}}$$ where $$||u||_{C(U)}:=\sup_{x \in U} |u(x)| \\ [u]_{C^{0, \gamma}}(U)=\sup_{x, y \in U , x \neq y} \frac{|u(x)-u(y)|}{|x-y|^{\gamma}}$$
So, we have the following:
$$||u_m-u_n||_{C^{0, \gamma}}=||u_m-u_n||_{C(U)}+[u_m-u_n]_{C^{0, \gamma}} \\ =\sup_{x \in U} |(u_m-u_n)(x)|+\sup_{x, y \in U , x \neq y} \frac{|(u_m-u_n)(x)-(u_m-u_n)(y)|}{|x-y|^{\gamma}} $$
How could we continue??