updated proof on diagonal intersection

inputs/lecture_14.tex

@@ -162,15 +162,17 @@ We have shown (assuming \AxC to choose contained clubs):
         We aim to show that $\gamma \in  \diagi_{\beta < \kappa} D_{\beta}$.
         Let $\beta_0 < \gamma$.
         We need to see that $\gamma \in D_{\beta_0}$.
+
         For each $\beta_0 \le \beta' < \gamma$
         there is some $\beta'' \in \diagi_{\beta < \kappa} D_\beta$
         such that $\beta' \le \beta'' < \gamma$,
         since $\gamma = \sup((\diagi_{\beta < \kappa} D_\beta) \cap \gamma)$.
         In particular $\beta'' \in D_{\beta_0}$.
 
-        We showed that $D_{\beta_0} \cap \gamma$
-        is unbounded in $\gamma$,
-        so $\gamma \in D_{\beta_0}$, since $D_{\beta_0}$ is closed.
+        So $D_{\beta_0} \cap \gamma$
+        is unbounded in $\gamma$.
+        Since $D_{\beta_0}$ is closed
+        it follows that $\gamma \in D_{\beta_0}$.
 
         %As $\beta_0 < \gamma$ was arbitrary,
         %this shows that $\gamma \in \diagi_{\beta < n} D_\beta$.