I am reading over the first answer for the question posted here: Holder Continuous Functions on $[0,1]$ are complete + Banach space
I am trying to understand some of the implicit details in the answer. The first answer says
- Let $(f_n)_{n=1}^\infty$ be Cauchy in $Λ_([0,1])$. You can use the Cauchy criterion to show $_()$ converges pointwise to some () for any ∈[0,1]
- $_→$ in the Hölder norm
For the sequence of functions to be Cauchy in $\Lambda_\alpha([0,1])$, is the meaning that for all $\varepsilon>0$, there exists an $N \in \mathbb{N}$ such that for all $m,n\geq N$ we have $$\lVert f_n - f_m \rVert < \varepsilon$$ according to the metric given in the space of Hölder continuous functions?
If this is right, then what the answer is saying is that we use this condition above to show that the sequence of functions converges pointwise to some $f$, but when we consider pointwise convergence, we are considering it in terms of the regular distance metric on $\mathbb{R}$ or $\mathbb{C}$?
Then is the point afterwards that once it is established that $f_n \to f$ pointwise in the distance metric, we need to check that it is also convergent in the Hölder norm? It was necessary to check that $f_n$ converged pointwise to ascertain the existence of an $f$?
And then the overall argument is proving that all Cauchy sequences converge to a point that is in the space, which is showing completeness of the space?
I just want to make sure if I am not misunderstanding anything about the process, and if these details are what is actually going on but isn't stated explicitly. Thanks.