A predefined proof environment provided by the amsthm package produces the heading ``Proof'' with appropriate spacing and punctuation. The proof environment is primarily intended for short proofs, no more than a page or two in length; longer proofs are usually better done as a separate section or subsection in your document.
A `Q.E.D.' symbol, , is automatically appended at the end of a proof environment. To substitute a different end-of-proof symbol, use renewcommand to redefine the command qedsymbol. For a long proof done as a subsection or section instead of with the proof environment, you can obtain the symbol and the usual amount of preceding space by using qed.
Placement of the Q.E.D. symbol can be problematic if the last part of a proof environment is a displayed equation or list environment or something of that nature. Adequate results can sometimes be obtained by using qed at the appropriate spot and then undefining qed just before the end of the proof. (The effect will be automatically localized to the current proof by normal / scoping rules.) For example:
\begin{proof}
...
\begin{equation}
G(t)=L\gamma!\,t^{-\gamma}+t^{-\delta}\eta(t) \qed
\end{equation}
\renewcommand{\qed}{}\end{proof}
An optional argument of the proof environment allows you to substitute a different name for the standard ``Proof''. If you want the proof heading to be, say, ``Proof of the Main Theorem'', then write
\begin{proof}[Proof of the Main Theorem]