In the linked question: Holder Continuous Functions on $[0,1]$ are complete + Banach space OP is trying to show an $\alpha$-Holder space ($\Lambda_{\alpha}$) is a Banach space. The first answer reads proceeds in three steps: producing a limit $f$, showing that $f_n \rightarrow f$ in norm, and finally showing $f$ is in the space.
To produce $f$ the poster assumes there's a Cauchy sequence in $\Lambda_{\alpha}$ and then states there exists a pointwise limit $f$ of this cauchy sequence.
My question: If we show $f \in \Lambda_{\alpha}$ aren't we done? -- we've assumed that a cauchy sequence in $\Lambda_{\alpha}$ and proved the limit lives in $\Lambda_{\alpha}$. If so, why does the poster bother checking $f_n \rightarrow f$ in norm separately in step 1? They should be able to just skip to step 2, right?