updated proof on diagonal intersection

inputs/lecture_14.tex

@@ -161,18 +161,19 @@ We have shown (assuming \AxC to choose contained clubs):
         is unbounded in $\gamma$.
         We aim to show that $\gamma \in  \diagi_{\beta < \kappa} D_{\beta}$.
         Let $\beta_0 < \gamma$.
-        We need to see $\gamma \in D_{\beta_0}$.
-        For each $\beta'$ with $\beta < \beta' < \gamma$,
+        We need to see that $\gamma \in D_{\beta_0}$.
+        For each $\beta_0 \le \beta' < \gamma$
         there is some $\beta'' \in \diagi_{\beta < \kappa} D_\beta$
-        with $\beta'' \ge  \beta', \beta'' < \gamma$.
+        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 $\gamma \in D_{\beta_0}$.
+        so $\gamma \in D_{\beta_0}$, since $D_{\beta_0}$ is closed.
 
-        As $\beta_0 < \gamma$ was arbitrary,
-        this shows that $\gamma \in \diagi_{\beta < n} D_\beta$.
+        %As $\beta_0 < \gamma$ was arbitrary,
+        %this shows that $\gamma \in \diagi_{\beta < n} D_\beta$.
     \end{subproof}
 
     \begin{claim}