Environments theorem and proof with enumerate

Question

How I can reproduce exactly the following text using \begin{proposition} \begin{enumerate} \item \item \end{enumerate} \end{proposition}

\begin{proof} \begin{enumerate} \item \item \end{enumerate} \end{proof}

Answer 1

You already have an exact reproduction of the text: just do a photocopy.

If you want to imitate that style, here's a possibility. I hope you don't want to exactly reproduce the errors I found, that are described in the proof.

\documentclass[a4paper]{book}
\usepackage{amsmath,amssymb,amsthm}

\usepackage{xpatch}
\xpatchcmd{\proof}{\itshape}{\scshape}{}{}

\usepackage{thmtools}
\usepackage{enumitem}


\declaretheoremstyle[
  notefont=\bfseries,
  bodyfont=\itshape,
]{hugo}

\declaretheorem[
  style=hugo,
  within=section
]{proposition}

\setlist[enumerate,1]{
  label=\upshape\arabic*.,
  ref=\arabic*,
  topsep=1ex,
  parsep=0pt,
}

\begin{document}
\setcounter{chapter}{3}
\setcounter{section}{8}
\setcounter{proposition}{2}

\begin{proposition}[Spectrum: finite-dimensional case]\label{prop-sfdc}
Let $X$ be a finite-dimensional normed space over $\mathcal{K}$,
with $\dim X=n$. Then
\begin{enumerate}
\item\label{prop-sfdc-x} Something.
\item\label{prop-sfdc-y} If $\mathcal{K}=\mathbb{C}$ something.
\item\label{prop-sfdc-z} Again.
\end{enumerate}
\end{proposition}

\begin{proof}
\ref{prop-sfdc-x}. Bla bla, but note \verb|\not\in| is wrong because it gives 
$\lambda\not\in\sigma(T)$, so \verb|\notin| should be used
$\lambda\notin\sigma(T)$.

\ref{prop-sfdc-y}. Note \verb|$U:X\rightarrow Y$| is wrong,
because it gives $U:X\to Y$, so \verb|\colon| should be used instead
of the colon, giving $U\colon X\to Y$.

\ref{prop-sfdc-z}. This ends the proof.
\end{proof}

The QED symbol at a double quad from the last word makes no sense 
whatsoever; either it is nearer to the last word or it is at the
far end of the line.

\end{document}

There is no enumerate in the proof: it's bad style doing that, even if the statement is divided into cases, because it produces long parts of indented text that can confuse readers, rather than helping them.

Answer 2

\documentclass{article}
\usepackage{amsthm,amsmath}
\usepackage{enumitem}
\newtheorem{proposition}[section]{Proposition}
\renewcommand{\proofname}{\scshape Proof}
\begin{document}
  \begin{proposition}[Spectral case]
    Let $X$ be a finite dimensional normed space over $\mathcal{K}$. Then
    \begin{enumerate}[label=\textup{\arabic*.},nosep]
      \item Blablabla
      \item more bla here
    \end{enumerate}
  \end{proposition}
  \begin{proof}
    \begin{enumerate}[nosep]
      \item Here comes the proof.
      \item And here more proof.
    \end{enumerate}
    Hence it is proved that some times, even if you show no research effort, we provide answers here. But please don't try your luck always
  \end{proof}
\end{document}