typo fix and add extra contents

pages/Math4111/Math4111_L2.md

@@ -61,31 +61,31 @@ WLOG $\alpha>\beta$ and $\beta>\alpha$.
 
 EOP
 
-We write $SupE$ to denote the LUB of $E$.
+We write $\sup E$ to denote the LUB of $E$.
 
 This also applies to $GLB$ (greatest lower bound) and infinum of $E$
 
-#### Example 
+Example:
 
 1. $S=\mathbb{Q}, E=\{1,2,3\}$ ($E$ is bounded above)
-    * $SupE=3$, $Inf E=1$
+    * $\sup E=3$, $\inf E=1$
 2. $S=\mathbb{Q}, E=\{x\in \mathbb{Q}:0<x<1\}$  ($E$ is bounded above)
-    * $SupE=3$, $Inf E=1$
+    * $\sup E=3$, $\inf E=1$
 
-$SupE$ and $Inf E=1$ don't have to $\in E$
+$\sup E$ and $\inf E=1$ don't have to $\in E$
 
 3. $S=\mathbb{Q}, E=\{x\in \mathbb{Q}:0<x\}$ ($E$ is not bounded above)
-   * $SupE=\infty$ or not defined, $Inf E=0$
+   * $\sup E=\infty$ or not defined, $\inf E=0$
 4. $S=\mathbb{Q}, E=\phi$.
-   * $SupE=-\infty$ or not defined, $Inf E=\infty$ or not defined, we don't put $\infty$ in $\mathbb{Q}$
+   * $\sup E=-\infty$ or not defined, $\inf E=\infty$ or not defined, we don't put $\infty$ in $\mathbb{Q}$
 
 Important example
 
 5. $S=\mathbb{Q}, A=\{p\leq \mathbb{Q}:p>0, p^2<2\}$.
-    * $A$ is not empty and bounded above. However, $Sup A$ des not exists.
+    * $A$ is not empty and bounded above. However, $\sup A$ des not exists.
 
 If $S=\mathbb{R}, A=\{p\leq \mathbb{Q}:p>0, p^2<2\}$.
-    * $A$ is not empty and bounded above. However, $Sup A=\sqrt{2}$.
+    * $A$ is not empty and bounded above. However, $\sup A=\sqrt{2}$.
 
 #### Least upper bound property (LUBP)
 
@@ -93,7 +93,7 @@ if $\forall E\subset S$ that tis non-empty and bounded above, $\exist Sup E\in S
 
 #### Greatest upper bound property (GLBP)
 
-S has greatest lower bound property (GLBP) if $\exist E\subset S$ that is non-empty and bounded below, $\exists Inf E\in S$
+S has greatest lower bound property (GLBP) if $\exist E\subset S$ that is non-empty and bounded below, $\exists \inf E\in S$
 
 $\mathbb{Q}$ does not have LUBP and GLBP.