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+# Lecture 3
+
+## Review
+
+Let $S=\mathbb{Z}$.
+
+1. Let $E=\{x\in S:x>0,x^2<5\}$. What are $sup\ E$ and $inf\ E$?
+  
+    $sup\ E=2,inf\ E=1$
+
+2. Can you find a subset $E\subset S$ which is bounded above but not bounded below?
+
+    $E=\{x\in S:x<0\}$
+
+3. Does $S$ have the least upper bound property?
+
+    Yes, $\forall E\subset S$ that tis non-empty and bounded above, $\exist Sup E\in S$.
+
+4. Does $S$ have the greatest lower bound property?
+
+    Yes, $\forall E\subset S$ that tis non-empty and bounded below, $\exist Inf E\in S$.
+
+## Continue
+
+### LUBP
+
+Proof that $LUBP\implies GLBP$.
+
+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.
+
+1. To show $L\neq \phi$.
+
+    $B$ is bounded below $\implies L\neq\phi$
+2. To show $L$ id bounded above.
+
+    $B$ 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 $S$).
+
+Let's say $\alpha=sup\ L$. We claim that $\alpha=inf\ B$. We need to show $2$ things.
+
+1. To show $\alpha$ is a lower bound of $B$, $\forall \gamma\in B,\alpha\leq \gamma$.
+
+    Let $\gamma\in B$, then $\gamma$ is an upper bound of $L$.
+
+    Since $\alpha$ is the least upper bound of $L$, $\alpha\leq \gamma$.
+
+2. To show $\alpha$ is the greatest lower bound of $B$, $\forall \beta>\alpha,\beta$ is not a lower bound of $B$.
+
+    Let $\beta>\alpha$. Since $\alpha$ is an upper bound of $L$, $\beta\notin L$.
+
+    By definition of $L$, $\beta$ is not a lower bound of $B$.
+
+Thus $\alpha=inf\ B$
+
+### Field
+
+|                | addition                                                    | multiplication                                                 |
+| -------------- | ----------------------------------------------------------- | -------------------------------------------------------------- |
+| closure        | $\checkmark$                                                | $\checkmark$                                                   |
+| commutativity  | $\checkmark$                                                | $\checkmark$                                                   |
+| associativity  | $\checkmark$                                                | $\checkmark$                                                   |
+| identity       | $\checkmark$ (denoted $0$)                                  | $\checkmark$ (denoted $1$)                                     |
+| inverses       | $\checkmark$ (denoted $-x$)                                 | $\checkmark$ (exists when $x\neq 0$ denoted $1/x$ or $x^{-1}$) |
+| distributivity | $\checkmark$ (distributive of multiplication over addition) ||
+
+Examples: $\mathbb{Q},\mathbb{R},\mathbb{C}$
+
+Non-examples: $\mathbb{N}$ fails A4,A5,M5, $\mathbb{Z}$ fails M5
+
+Another example of field: $\mathbb{Z}/5\mathbb{Z}=\{1,2,3,4,5\}$, $\forall a,b\in \mathbb{Z}/5\mathbb{Z}$, $a+b=(a+b)\mod 5$, $a\cdot b=(a\cdot b)\mod 5$
+
+Some properties of fields: see Proposition 1.14,1.15,1.16
+
+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: 
+
+#### 1.14(a) $x+y=x+z\implies y=z$
+
+Proof: 
+
+$x+y=x+z$,
+
+$(-x)+(x+y)=(-x)+(x+z)$,
+
+by A3, $(-x+x)+(y)=(-x+x)+(z)$,
+
+$0+y=0+z$,
+
+$y=z$.
+
+Chain of equalities.
+
+#### 1.16(a) $\forall x\in \mathbb{F}, 0x=0$
+
+1. A4, where 0 is defined.
+2. Since $0$ is defined in the addition, identity. The proposition says something about multiplication by 0. The only proposition that relates the addition and multiplication is Distributive law.
+
+$0x=(0+0)x=0x+0x$, cancel $0x$ on both side we have $0x=0$.
+
+### Ordered Field (1.17)
+
+An _ordered field_ is a _field_ $F$ which is also an _ordered set_, such that
+
+1. $x+y<x+z$ if $x,y,\in F$ and $y<z$,
+2. $xy>0$ if $x\in F,y\in F,x>0$ and $y>0$.
+
+#### Prop 1.18
+
+If $x>0$ and $y<z$, then $xy<yz$.
+
+Proof: $y<z\implies 0<z-y$, $x(z-y)>0\implies xz>xy$
+
+We define $\mathbb{R}$ to be the unique ordered field with $LUBP$. (The existence and uniqueness are discussed in the appendix of this chapter).
+
+#### Theorem 1.20
+
+1. (Archimedean property) If $x,y\in \mathbb{R}$ and $x>0$, then $\exists n\in \mathbb{N}$ such that $nx>y$.
+2. ($\mathbb{Q}$ is dense in $\mathbb{R}$) If $x,y\in \mathbb{R}$ and $x<y$, then $\exists p\in \mathbb{Q}$$ such that $x<p<y$.