Let $f_n:[a,b] \rightarrow R$ continous Hölder functions such that
$$|f_n|_\alpha=\sup_{x\neq y} \frac{|f_n(x)-f_n(y)|}{|x-y|^{\alpha}}\leq M$$ for every $n\in N$. Prove that there exists a subsequence which converges uniformly to a Hölder function $f$.
We must show that this family of functions are equibounded and equicontinuous. What I have done so far was to prove equicontinuity as follows:
Since $|f_n|_\alpha\leq M$, we have $|f_n(x)-f_n(y)|\leq M|x-y|^{\alpha}, x\neq y.$ Let $\epsilon>0$ be given. Choose $\delta= (\epsilon/M)^{1/\alpha}. $ If $|x-y|<\delta$, then by the above condition we have $|f_n(x)-f_n(y)|<\epsilon.$ The family is equicontinuous.
But I cannot prove that it is equibounded. Any help or hint? Thank you.
You can't prove it because the hypothesis is missing. Namely, the sequence $f_n(x)=n+x$ satisfies the hypothesis of the exercise for $\alpha=1$, but it isn't equibounded, nor does it satisfy the thesis.