diff --git a/ws2019/ana/lectures/analysis.pdf b/ws2019/ana/lectures/analysis.pdf
index 527f985..e539f5c 100644
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diff --git a/ws2019/ana/lectures/analysis.tex b/ws2019/ana/lectures/analysis.tex
index d3e1cce..bc30ab5 100644
--- a/ws2019/ana/lectures/analysis.tex
+++ b/ws2019/ana/lectures/analysis.tex
@@ -22,7 +22,6 @@
 \input{analysis12.tex}
 \input{analysis13.tex}
 \input{analysis14.tex}
-\input{analysis14.tex}
 \input{analysis15.tex}
 
 \end{document}
diff --git a/ws2019/ana/lectures/analysis15.pdf b/ws2019/ana/lectures/analysis15.pdf
index 481d962..9ab150b 100644
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diff --git a/ws2019/ana/lectures/analysis15.tex b/ws2019/ana/lectures/analysis15.tex
index e8bd302..9b3859c 100644
--- a/ws2019/ana/lectures/analysis15.tex
+++ b/ws2019/ana/lectures/analysis15.tex
@@ -22,7 +22,7 @@
 \subsection{Grenzwerte bei Funktionen}
 
 \begin{definition}[Berührpunkt]
-    Sei $D \in \R$. Ein Punkt $a \in \R$ heißt Berührpunkt von $D$, falls
+    Sei $D \subset  \R$. Ein Punkt $a \in \R$ heißt Berührpunkt von $D$, falls
     in jeder $\delta$-Umgebung von $a$, d.h.
     \[
         U_{\delta}(a) := ]a - \delta, a + \delta[ = (a-\delta, a+\delta)