small changes

inputs/lecture_07.tex

@@ -43,8 +43,8 @@ For the proof we'll need a slight generalization of \autoref{thm2}:
     By \autoref{thm4} it follows that $\sum_{n \ge 1}  Y_n < \infty$
     almost surely.
     Let $A_n \coloneqq \{\omega : |X_n(\omega)| > C\}$.
-    Since the first series $\sum_{n \ge 1}  \bP(A_n) < \infty$,
-    by Borel-Cantelli, $\bP[\text{infinitely many $A_n$ occur}] = 0$.
+    Since $\sum_{n \ge 1}  \bP(A_n) < \infty$ by assumption,
+    Borel-Cantelli yields $\bP[\text{infinitely many $A_n$ occur}] = 0$.
 
 
     For the proof of (b), suppose $\sum_{n\ge 1} X_n(\omega) < \infty$