[Tex/LaTex] enumerated list in proof – margin, qed symbol

Question

I have two questions.
1. Can I delete the red-colored space?
2. If I use enumerate in proof, QED symbol appears the next line of the end, as above. If I do not want to type manually \qedhere, should I use ntheorem?

\documentclass[a4paper]{article}
\usepackage[scale=0.75]{geometry}

\usepackage[english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{textcomp}
\usepackage{enumitem}
\setlist[enumerate]{nosep, label=(\arabic*)}

\theoremstyle{definition}
\newtheorem{lem}{Lemma}

\begin{document}

\begin{lem}
\begin{enumerate}
\item xxxx
\item yyyy
\end{enumerate}
\end{lem}
\begin{proof}
\begin{enumerate}
\item xxxxxxxxxxxxxxxxxxxxxxxxx. xxxxxxxxxxxxxxxxxxxxxxxxx. xxxxxxxxxxxxxxxxxxxxxxxxx. xxxxxxxxxxxxxxxxxxxxxxxxx. xxxxxxxxxxxxxxxxxxxxxxxxx. xxxxxxxxxxxxxxxxxxxxxxxxx. 
\item yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. \end{enumerate}
\end{proof}

\end{document}

Answer 1

amsthm already has the mechanism, called \qedhere:

\documentclass[a4paper]{article}

\usepackage[english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{textcomp}
\usepackage{enumitem}
\setlist[enumerate]{nosep, label=(\arabic*)}

\theoremstyle{definition}
\newtheorem{lem}{Lemma}

\begin{document}

\begin{lem}
\begin{enumerate}
\item xxxx
\item yyyy
\end{enumerate}
\end{lem}
\begin{proof}
\begin{enumerate}

\item xxxx xxx xxxxxx xxxxx xxxxxx. xxxxx xxx xxxxxx xxxxx xxxxxx. xxxxx xxx xxxxxx xxxxx 
xxxxxx. xxxxx xxx xxxxxx xxxxx xxxxxx. xxxxx xxx xxxxxx xxxxx xxxxxx. xxxxx xxx xxxxxx xxxxx 
xxxxxx.

\item yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy 
yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. 
yyyyy yyy yyyy. yyyyy yyy yyyy.\qedhere
\end{enumerate}

\end{proof}

\end{document}

You can use it also at the end of displayed equations.

If you want to remove the indent, just don't use enumerate.

\documentclass[a4paper]{article}

\usepackage[english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{textcomp}
\usepackage{enumitem}
\setlist[enumerate]{nosep, label=(\arabic*)}

\theoremstyle{definition}
\newtheorem{lem}{Lemma}

\begin{document}

\begin{lem}
\begin{enumerate}
\item xxxx
\item yyyy
\end{enumerate}
\end{lem}
\begin{proof}
(1) xxxx xxx xxxxxx xxxxx xxxxxx. xxxxx xxx xxxxxx xxxxx xxxxxx. xxxxx xxx xxxxxx xxxxx 
xxxxxx. xxxxx xxx xxxxxx xxxxx xxxxxx. xxxxx xxx xxxxxx xxxxx xxxxxx. xxxxx xxx xxxxxx xxxxx 
xxxxxx.

(2) yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy 
yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. 
yyyyy yyy yyyy. yyyyy yyy yyyy.
\end{proof}

\end{document}