From 901f9140d6b55787fd4ac80b9808431d0e143b20 Mon Sep 17 00:00:00 2001
From: Frederik van der Els <49305700+xpple@users.noreply.github.com>
Date: Thu, 16 Jan 2025 20:58:49 +0100
Subject: [PATCH] Fix limit definition

---
 content/history/set-theory/limits.tex | 10 +++++++++-
 1 file changed, 9 insertions(+), 1 deletion(-)

diff --git a/content/history/set-theory/limits.tex b/content/history/set-theory/limits.tex
index 02dbf8cb..4944f02a 100644
--- a/content/history/set-theory/limits.tex
+++ b/content/history/set-theory/limits.tex
@@ -34,8 +34,16 @@
 making $x$ suitably close to~$c$. More precisely, we stipulate that
 $\lim_{x \rightarrow c} g(x) = \ell$ will mean:
 \[
-(\forall\epsilon > 0)(\exists \delta > 0)\forall x \left(|x - c| < \delta \lif |g(x) - \ell| < \epsilon \right).
+(\exists\ell \in \Real)(\forall\epsilon > 0)(\exists \delta > 0)\forall x \left(0 < |x - c| < \delta \lif |g(x) - \ell| < \epsilon \right).
 \]
+Here $\ell$ is unique. That is, if both $\ell_1 \in \Real$ and
+$\ell_2 \in \Real$ witness the above sentence, it must be the case
+that $\ell_1=\ell_2$. We say that the limit of a function is unique.
+
+\begin{prob}
+	Prove that the limit of a function is unique.
+\end{prob}
+
 The vertical bars here indicate absolute magnitude. That is, $|x| = x$
 when $x \geq 0$, and $|x| =-x$ when $x < 0$; you can depict
 \emph{that} function as follows: