diff --git a/ann/ann_notes.pdf b/ann/ann_notes.pdf
index 1b1ee27..7cc9a08 100644
Binary files a/ann/ann_notes.pdf and b/ann/ann_notes.pdf differ
diff --git a/ann/ann_notes.tex b/ann/ann_notes.tex
index af86d63..a26f21c 100644
--- a/ann/ann_notes.tex
+++ b/ann/ann_notes.tex
@@ -3378,7 +3378,7 @@
           Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function and
           $a \in \mathcal{D}_f$. The $f$ function is differentiable at the point
           $a$ if an $I \subset \mathcal{D}_f$ non-empty open interval and an
-          $L \in \mathbb{R}$ number exists such that $a \in I$ and
+          $L \in \mathbb{R}$ number exist such that $a \in I$ and
 
           $$
             \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} = L