diff --git a/ia/analysis/10_integration.tex b/ia/analysis/10_integration.tex
index 3194abe..975cb23 100644
--- a/ia/analysis/10_integration.tex
+++ b/ia/analysis/10_integration.tex
@@ -151,7 +151,10 @@ \subsection{Monotonic and continuous functions}
 \end{theorem}
 \begin{proof}
 	Suppose \(f\) is increasing.
-	Then \(\sup_{x \in [x_{j-1} - x_j]} f(x) = f(x_j)\), and similarly, \(\inf_{x \in [x_{j-1} - x_j]} f(x) = f(x_{j-1})\).
+	Then
+	\[\sup_{x \in [x_{j-1} - x_j]} f(x) = f(x_j)\]
+	and similarly
+	\[\inf_{x \in [x_{j-1} - x_j]} f(x) = f(x_{j-1})\]
 	Thus,
 	\[
 		S(f, \mathcal D) - s(f, \mathcal D) = \sum_{j=1}^n (x_j - x_{j-1}) \left[ f(x_j) - f(x_{j-1}) \right]
@@ -202,7 +205,7 @@ \subsection{Monotonic and continuous functions}
 		\mathcal D = \qty{ a, a + \frac{b-a}{n}, a + 2\frac{b-a}{n} + \dots + b }
 	\]
 	where \(n\) is chosen large enough such that \(\frac{b-a}{n} < \delta\).
-	Then, for any \(x, y \in [x_{j-1}, x_j]\), \(\abs{f(x) - f(y)} < \varepsilon\).
+	Then, for any \(x, y \in [x_{j-1}, x_j]\), we have that \(\abs{f(x) - f(y)} < \varepsilon\).
 	We can now write
 	\[
 		\max_{x \in [x_{j-1}, x_j]} f(x) - \min_{x \in [x_{j-1}, x_j]} f(x) = f(p) - f(q) < \varepsilon
diff --git a/ib/opt/04_linear_programming.tex b/ib/opt/04_linear_programming.tex
index 4baaba8..e2c9dca 100644
--- a/ib/opt/04_linear_programming.tex
+++ b/ib/opt/04_linear_programming.tex
@@ -146,7 +146,7 @@ \subsection{Basic solutions and basic feasible solutions}
 \begin{enumerate}[A:]
 	\setcounter{enumi}{2}
 	\item Every basic solution has \textit{exactly} \( m \) nonzero entries.
-	      This is known as the non-degeneracy assumption.
+	      This assumption is known as the non-degeneracy assumption.
 	      This assumption cannot be created without loss of generality, but it is far simpler to discuss problems with this assumption met.
 	      Throughout this course, we will keep this assumption to be true.
 \end{enumerate}
diff --git a/ib/vp/04_extensions_to_the_euler_lagrange_equation.tex b/ib/vp/04_extensions_to_the_euler_lagrange_equation.tex
index d4bffad..8738cbd 100644
--- a/ib/vp/04_extensions_to_the_euler_lagrange_equation.tex
+++ b/ib/vp/04_extensions_to_the_euler_lagrange_equation.tex
@@ -122,7 +122,7 @@ \subsection{Geodesics on surfaces}
 	\Sigma = \qty{ \vb x \colon g(\vb x) = 0 }
 \]
 Consider two points \( A, B \) on \( \Sigma \).
-What are the geodesics (shortest paths on the surface) between the two points, if one exists at all?
+What are the geodesics (the shortest paths on the surface) between the two points, if one exists at all?
 Consider a parametrisation of such a path given by \( t \in [0, 1] \) where \( A = \vb x(0), B = \vb x(1) \).
 We wish to extremise
 \[