Typos

Advanced/Proof Techniques/parts/0 intro.tex

@@ -75,7 +75,7 @@ This is a trick we often use when showing that two quantities are equal.
 
 
 \problem{}
-Although $A \iff B$ looks like a single statement, we often need to prove each direction seperately. \par
+Although $A \iff B$ looks like a single statement, we often need to prove each direction separately. \par
 Show that $x \in \mathbb{Z}$ iff $\lfloor x \rfloor = \lceil x \rceil$
 
 \begin{solution}
@@ -97,7 +97,7 @@ Show that $x \in \mathbb{Z}$ iff $\lfloor x \rfloor = \lceil x \rceil$
 \pagebreak
 
 \problem{}
-We don't always need to prove each direction of an iff statement seperately. \par
+We don't always need to prove each direction of an iff statement separately. \par
 
 \begin{itemize}[itemsep = 1mm]
 	\item Convince yourself that we can \say{chain} iffs together: \par

Advanced/Proof Techniques/parts/1 contradiction.tex

@@ -1,8 +1,8 @@
 \section{Proofs by Contradiction}
 
 \definition{}
-A very common (and somewhat contraversial) proof technique is
-\textit{proof by contradiction}. It works as follows: 
+A very common proof technique is \textit{proof by contradiction}.
+It works as follows: 
 
 \vspace{2mm}
 
@@ -18,7 +18,7 @@ Show that the set of integers has no maximum using a proof by contradiction.
 \begin{solution}
 	Assume there is a maximal integer $x$. \par
 	$x + 1$ is also an integer. \par
-	$x + 1$ is larger than $x$, which contradicts our original assumtion!
+	$x + 1$ is larger than $x$, which contradicts our original assumption!
 
 	\vspace{2mm}
 

Advanced/Proof Techniques/parts/2 induction.tex

@@ -229,7 +229,7 @@ For example, see the proof of the statement in \ref{binomsum} on the next page.
 	\textbf{Solution 2:}\par
 	We could also observe that there are $x - 1$ places to put a \say{bar} in
 	the array of ones. This corresponds to $x - 1$ binary positions, and thus
-	$2^{x-1}$ ways to seperate our array of $1$s with bars.
+	$2^{x-1}$ ways to separate our array of $1$s with bars.
 
 	\linehack{}