diff --git a/inputs/lecture_14.tex b/inputs/lecture_14.tex
index e7d2ce2..59914fa 100644
--- a/inputs/lecture_14.tex
+++ b/inputs/lecture_14.tex
@@ -153,7 +153,7 @@ Let $\rho(\prec) \coloneqq  \sup \{\rho_{\prec}(x) + 1 : x \in X\}$.
 
 \begin{fact}[{\cite[Appendix B]{kechris}}]
     Since $\rho_\prec\colon X \to \rho(\prec)$ is surjective,
-    we have that $\rho(\prec) \le  |X|^+$.%
+    we have that $\rho(\prec) <  |X|^+$.%
     \gist{\footnote{Here, $|X|^+$ denotes the successor cardinal.}}{}
 \end{fact}
 \begin{theorem}[{Kunen-Martin, \cite[(31.1)]{kechris}}]
diff --git a/inputs/tutorial_08.tex b/inputs/tutorial_08.tex
index c6ccca4..fd1c1f3 100644
--- a/inputs/tutorial_08.tex
+++ b/inputs/tutorial_08.tex
@@ -265,20 +265,20 @@ $f$ and $g$ are bijections between
 $X_\omega \coloneqq \bigcap X_i$ and $Y_\omega \coloneqq \bigcap Y_i$.
 
 
-% https://q.uiver.app/#q=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
+% https://q.uiver.app/#q=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
 \adjustbox{scale=0.7,center}{%
-\begin{tikzcd}
-	{X \setminus X_\omega =} & {(X_0 \setminus X_1)} & \cup & {(X_0 \setminus X_1)} & \cup & {(X_0 \setminus X_1)} & \cdots & {} \\
-	{Y\setminus Y_\omega =} & {(Y_0 \setminus Y_1)} & \cup & {(Y_0 \setminus Y_1)} & \cup & {(Y_0 \setminus Y_1)} & \cdots & {}
+\[\begin{tikzcd}
+	{X \setminus X_\omega =} & {(X_0 \setminus X_1)} & \cup & {(X_1 \setminus X_2)} & \cup & {(X_2 \setminus X_3)} & \cdots & {} \\
+	{Y\setminus Y_\omega =} & {(Y_0 \setminus Y_1)} & \cup & {(Y_1 \setminus Y_2)} & \cup & {(Y_2 \setminus Y_3)} & \cdots & {}
 	\arrow["f"'{pos=0.7}, from=1-2, to=2-4]
 	\arrow["g"{pos=0.1}, from=2-2, to=1-4]
 	\arrow["f"{pos=0.8}, from=1-6, to=2-8]
 	\arrow["g"{pos=0.1}, from=2-6, to=1-8]
-\end{tikzcd}
+\end{tikzcd}\]
 }
 
 By \autoref{thm:lusinsouslin}
-the injective image via a Borel set of a Borel set is Borel.
+the injective image via a Borel function of a Borel set is Borel.
 
 \autoref{thm:lusinsouslin} also gives that the inverse
 of a bijective Borel map is Borel.