Merge pull request #5 from DeMorgan93/tiny-fixes

Typos and small proof extension

Author: FabsOliveira

src/chapters/chapter_2/chapter_1-2.tex

@@ -509,7 +509,7 @@ \section{Extreme points, vertices, and basic feasible solutions}
 			\begin{equation*}
 				c^\top x = \sum_{i \in I} a_i^\top x \geq \sum_{i \in I} b_i, 	 	
 	 		\end{equation*}
-	 		since $a_i^\top x \geq b_i$ for $i \in M_1 \cup M_2$. Thus, for any $x \in P$, $c^\top \overline{x} \leq c^\top x$, making $\overline{x}$ a vertex (cf. Definition \ref{p1c2:def:vertex}). \qedhere
+	 		since $a_i^\top x \geq b_i$ for $i \in M_1 \cup M_2$. Thus, for any $x \in P$, $c^\top \overline{x} \leq c^\top x$. The equality $c^\top \overline{x} = c^\top x$ holds if $a_i^\top x = b_i$ for $i \in I$. However, $\overline{x}$ is a BFS and thus a unique solution to this system of equations. Therefore, $c^\top \overline{x} < c^\top x$, making $\overline{x}$ a vertex (cf. Definition \ref{p1c2:def:vertex}). \qedhere
 	\end{enumerate} 	
 \end{proof}