Minor edits

Advanced/Nonstandard Analysis/parts/1 extensions.tex

@@ -116,7 +116,7 @@ Then, show that a negative $\delta$ is infinitesimal if and only if it is bigger
 Prove the following statements: \par
 \begin{itemize}
 	\item If $\delta$ and $\varepsilon$ are infinitesimal, then $\delta + \varepsilon$ is infinitesimal.
-	\item If $\delta$ is infinitesimal and $x$ is limited, then $a\delta$ is infinitesimal.
+	\item If $\delta$ is infinitesimal and $x$ is limited, then $x\delta$ is infinitesimal.
 	\item If $x$ and $y$ are limited, $xy$ and $x+y$ are too.
 	\item A nonzero $\delta$ is infinitesimal iff $\delta^{-1}$ is unlimited.
 \end{itemize}
@@ -126,9 +126,9 @@ Prove the following statements: \par
 \problem{}
 Let $\delta$ be a positive infinitesimal. Which is greater?
 \begin{itemize}
-	\item $\delta$ or $\delta^2$?
-	\item $(1 - \delta)$ or $(1 + \delta^2)^{-1!}$?
-	\item $\frac{1 + \delta}{1 + \delta^2}$ or $\frac{2 + \delta^2}{2 + \delta^3}$? \par
+	\item $\delta$ or $\delta^2$
+	\item $(1 - \delta)$ or $(1 + \delta^2)^{-1!}$
+	\item $\frac{1 + \delta}{1 + \delta^2}$ or $\frac{2 + \delta^2}{2 + \delta^3}$ \par
 	\note[Note]{we define $\frac{1}{x}$ as $x^{-1}$, and thus $\frac{a}{b} = a \times b^{-1}$}
 \end{itemize}
 
@@ -138,13 +138,13 @@ Let $\delta$ be a positive infinitesimal. Which is greater?
 
 \definition{}
 We say two elements of an ordered field are \textit{infinitely close} if $x - y$ is infinitesimal. \par
-We say that $x_0 \in \mathbb{R}$ is a \textit{standard part} of $x$ if it is infinitely close to $x$. \par
+We say that $x_0 \in \mathbb{R}$ is the \textit{standard part} of $x$ if it is infinitely close to $x$. \par
 
 \problem{}
 We will denote the standard part of $x$ as $\text{st}(x)$. \par
 Show that $\text{st}(x)$ is well-defined for limited $x$. \par
-(In other words, Show that $x_0$ exists and is unique for limited $x$). \par
-\hint{To prove existance, consider $\text{sup}(\{a \in \mathbb{R} ~|~ a < x\}$)}
+(In other words, show that $x_0$ exists and is unique for limited $x$). \par
+\hint{To prove existence, consider $\text{sup}(\{a \in \mathbb{R} ~|~ a < x\}$)}
 
 \vfill
 

Advanced/Nonstandard Analysis/parts/2 dual.tex

@@ -58,7 +58,7 @@ What is the derivative of $f(x) = x^n$?
 \vfill
 
 \problem{}
-Say the derivatives of $f$ and $g$ are known. \par
+Assume that the derivatives of $f$ and $g$ are known. \par
 Find the derivatives of $h(x) = f(x) + g(x)$ and $k(x) = f(x) \times g(x)$.
 
 \vfill
@@ -101,7 +101,7 @@ Find the derivative of the following functions:
 \vfill
 
 \problem{}
-Say the derivatives of $f$ and $g$ are known. \par
+Assume that the derivatives of $f$ and $g$ are known. \par
 What is the derivative of $f(g(x))$?
 
 \vfill