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.
 

pages/Math4111/Math4111_L3.md

@@ -4,7 +4,7 @@
 
 Let $S=\mathbb{Z}$.
 
-1. Let $E=\{x\in S:x>0,x^2<5\}$. What are $sup\ E$ and $inf\ E$?
+1. Let $E=\{x\in S:x>0,x^2<5\}$. What are $sup\ E$ and $\inf\ E$?
   
     $sup\ E=2,inf\ E=1$
 
@@ -14,11 +14,11 @@ Let $S=\mathbb{Z}$.
 
 3. Does $S$ have the least upper bound property?
 
-    Yes, $\forall E\subset S$ that tis non-empty and bounded above, $\exist Sup E\in S$.
+    Yes, $\forall E\subset S$ that tis non-empty and bounded above, $\exist \sup E\in S$.
 
 4. Does $S$ have the greatest lower bound property?
 
-    Yes, $\forall E\subset S$ that tis non-empty and bounded below, $\exist Inf E\in S$.
+    Yes, $\forall E\subset S$ that tis non-empty and bounded below, $\exist \inf E\in S$.
 
 ## Continue
 
@@ -26,9 +26,9 @@ Let $S=\mathbb{Z}$.
 
 Proof that $LUBP\implies GLBP$.
 
-Let $S$ be an ordered set with LUBP. Let B<S be non-empty and bounded below.
+Let $S$ be an ordered set with LUBP. Let $B<S$ be non-empty and bounded below.
 
-Let $L=y\in S:y$ is a lower bound of B$\}$. From the picture, we expect $sup\ L=inf\ B$ First we'll show $sup\ L$ exists.
+Let $L=y\in S:y$ is a lower bound of $B$. From the picture, we expect $\sup L=\inf B$ First we'll show $\sup L$ exists.
 
 1. To show $L\neq \phi$.
 
@@ -37,7 +37,7 @@ Let $L=y\in S:y$ is a lower bound of B$\}$. From the picture, we expect $sup\ L=
 
     $B$ is not empty $\implies \exists x\in B\implies x$ is a upper bound of $L$.
 
-3. Since $S$ has the least upper bound property, $sup L$ exists (in $S$).
+3. Since $S$ has the least upper bound property, $\sup L$ exists (in $S$).
 
 Let's say $\alpha=sup\ L$. We claim that $\alpha=inf\ B$. We need to show $2$ things.
 
@@ -79,11 +79,11 @@ Remark:
 1. It's more helpful if you try to prove these yourselves. The proofs are "straightforward".
 2. For this course, it's not important to remember which properties are axioms, etc.
 
-Example of proof: 
+Example of proof:
 
 #### 1.14(a) $x+y=x+z\implies y=z$
 
-Proof: 
+Proof:
 
 $x+y=x+z$,
 

pages/Math4111/Math4111_L4.md

@@ -2,7 +2,7 @@
 
 ## Review
 
-1. Let $F$ be a field. Let $a,b,c,...,z\in F$ . Using he field axioms, simplify 
+1. Let $F$ be a field. Let $a,b,c,...,z\in F$ . Using he field axioms, simplify
 
     $$
     (x-a)(x-b)(x-c)...(x-z)
@@ -10,7 +10,7 @@
 
     $x\in F$, it must be at least one $0$ in the product...
 
-2. Suppose $A,B\subset\mathbb{R}$. Suppose $A$ and $B$ are nonempty and bounded above,$A\subset B$. WHat can you say about $sup\ A$ and $sup\ B$? Please justify.
+2. Suppose $A,B\subset\mathbb{R}$. Suppose $A$ and $B$ are nonempty and bounded above,$A\subset B$. WHat can you say about $\sup A$ and $\sup B$? Please justify.
 
     $$
     \forall x\in A, x\in B.  sup\ A\leq sup\ B
@@ -31,7 +31,7 @@ Proof
 
 Suppose the property is false, then $\exist x,y\in \mathbb{R}$ with $x>0$ such that $\forall v\in \mathbb{N}$, nx\leq y$
 
-Let $A=\{nx:n\in\mathbb{N}\}$. Then $A\neq\phi$ (Since $x\in A$) and $A$ is bounded above by $y$. Since $\mathbb{R}$ has LUBP, $sup\ A$ exists. Let $\alpha=sup\ A$.
+Let $A=\{nx:n\in\mathbb{N}\}$. Then $A\neq\phi$ (Since $x\in A$) and $A$ is bounded above by $y$. Since $\mathbb{R}$ has LUBP, $sup\ A$ exists. Let $\alpha=\sup A$.
 
 $x>0\implies \alpha-x<\alpha$, $\alpha-x$ is not an upper bound of $A$. (Since $\alpha$ is the LUB of $A$) $\implies \exist m\in \mathbb{N}$ such that $mx>\alpha-x$ by definition of $A$.
 

pages/Math4111/Math4111_L8.md

@@ -10,7 +10,7 @@ It should be empty. Proof any point cannot be in two balls at the same time. (By
 
 ### Metric space defs
 
-1. $p\in X,r>0$, $B_r(p)=\{q\in X:d(p,q)<0\}$, also called **neighborhood**.
+1. $p\in X,r>0$, $B_r(p)=\{q\in X:d(p,q)<r\}$, also called **neighborhood**.
 2. $p$ is a **limit point** of $E(p\in E')$ if $\forall r>0$, $(B_s(p)\cap E)\backslash \{p\}\neq \phi$
 3. If $p\in E$ and $p$ is not a limit point of $E$, then $p$ is called an **isolated point** of $E$.
 4. $E$ is **closed** if $E'\subset E$