updates

content/Math4111/Math4111_L2.md

@@ -47,19 +47,18 @@ Let $S$ be an ordered set and $E\subset S$. We say $\alpha\in S$ is the LUB of $
 1. $\alpha$ is the UB of $E$. ($\forall x\in E,x\leq \alpha$)
 2. if $\gamma<\alpha$, then $\gamma$ is not UB of $E$. ($\forall \gamma <\alpha, \exist x\in E$ such that $x>\gamma$ )
 
-#### Lemma 
-
-Uniqueness of upper bounds.
+#### Lemma (Uniqueness of upper bounds)
 
 If $\alpha$ and $\beta$ are LUBs of $E$, then $\alpha=\beta$.
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 Suppose for contradiction $\alpha$ and $\beta$ are both LUB of $E$, then $\alpha\neq\beta$
 
 WLOG $\alpha>\beta$ and $\beta>\alpha$.
 
-QED
+</details>
 
 We write $\sup E$ to denote the LUB of $E$.