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typo fix and add extra contents

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2025-02-22 16:35:08 -06:00
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pages/Math4111/Math4111_L3.md
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@@ -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:
$x+y=x+z$,