update notations and fix typos

pages/Math4121/Math4121_L10.md

@@ -40,7 +40,7 @@ By **Theorem 6.11**, $f^2,g^2\in \mathscr{R}(\alpha)$ on $[a, b]$.
 
 By linearity, $fg=1/4((f+g)^2-(f-g)^2)\in \mathscr{R}(\alpha)$ on $[a, b]$.
 
-EOP
+QED
 
 (b) $|f|\in \mathscr{R}(\alpha)$ on $[a, b]$, and $|\int_a^b f d\alpha|\leq \int_a^b |f| d\alpha$.
 
@@ -54,7 +54,7 @@ Let $c=-1$ or $c=1$. such that $c\int_a^b f d\alpha=| \int_a^b f d\alpha|$.
 
 By linearity, $c\int_a^b f d\alpha=\int_a^b cfd\alpha$. Since $cf\leq |f|$, by monotonicity, $|\int_a^b cfd\alpha|=\int_a^b cfd\alpha\leq \int_a^b |f| d\alpha$.
 
-EOP
+QED
 
 ### Indicator Function
 
@@ -104,6 +104,6 @@ Since $f$ is continuous at $s$, when $x\to s$, $U(P,f,\alpha)\to f(s)$ and $L(P,
 
 Therefore, $U(P,f,\alpha)-L(P,f,\alpha)\to 0$, $f\in \mathscr{R}(\alpha)$ on $[a, b]$, and $\int_a^b f d\alpha=f(s)$.
 
-EOP
+QED
 
 

pages/Math4121/Math4121_L11.md

@@ -79,7 +79,7 @@ $$
 \end{aligned}
 $$
 
-EOP
+QED
 
 If $f\in \mathscr{R}$, and there exists a differentiable function $F:[a,b]\to \mathbb{R}$ such that $F'=f$ on $(a,b)$, then
 
@@ -107,4 +107,4 @@ $$
 
 So, $f\in \mathscr{R}$ and $\int_a^b f(x)\ dx=F(b)-F(a)$.
 
-EOP
+QED

pages/Math4121/Math4121_L14.md

@@ -26,7 +26,7 @@ Proof:
 
 To prove Riemann's Integrability Criterion, we need to show that a bounded function $f$ is Riemann integrable if and only if for every $\sigma, \epsilon > 0$, there exists a partition $P$ of $[a, b]$ such that the sum of the lengths of the intervals where the oscillation exceeds $\sigma$ is less than $\epsilon$.
 
-EOP
+QED
 
 #### Proposition 2.4
 

pages/Math4121/Math4121_L15.md

@@ -36,7 +36,7 @@ So $S\setminus \bigcup_{i=1}^{n} I_i$ contains only finitely many points, say $N
 
 So $c_e(S)\leq \ell(C)\leq c_e(S')+\epsilon$.
 
-EOP
+QED
 
 #### Corollary: sef of first species
 

pages/Math4121/Math4121_L16.md

@@ -32,7 +32,7 @@ So $d\in S$, this contradicts the definition of $\beta$ as the supremum of $S$.
 
 So $\beta \geq b$.
 
-EOP
+QED
 
 ### Reviewing sections for Math 4111
 
@@ -91,4 +91,4 @@ Setting $r=\sup a_n$ (by the least upper bound property of real numbers), $r\in
 
 This contradicts the assumption that $a_n,b_n$ as the first element in the list.
 
-EOP
+QED

pages/Math4121/Math4121_L17.md

@@ -38,7 +38,7 @@ If $b\in T$, then by definition of $T$, $b \notin \psi(b)$, but $\psi(b) = T$, w
 
 If $b \notin T$, then $b \in \psi(b)$, which is also a contradiction since $b\in T$. Therefore, $2^S$ cannot have the same cardinality as $S$.
 
-EOP
+QED
 
 ### Back to Hankel's Conjecture
 

pages/Math4121/Math4121_L2.md

@@ -39,7 +39,7 @@ $$
 
 So $h'(x)=\frac{h(t)-h(x)}{t-x}=(f'(x)+u(t))(g'(y)+v(s))$. Since $u(t)\to 0$ and $v(s)\to 0$ as $t\to x$ and $s\to y$, we have $h'(x)=g'(y)f'(x)$.
 
-EOP
+QED
 
 #### Example 5.6
 
@@ -135,4 +135,4 @@ If $x<t<x+\delta$, then $f(x)\geq f(t)$ so $\frac{f(t)-f(x)}{t-x}\geq 0$.
 
 So $\lim_{t\to x}\frac{f(t)-f(x)}{t-x}=0$.
 
-EOP
+QED

pages/Math4121/Math4121_L3.md

@@ -39,7 +39,7 @@ Consider three cases:
 
 In all cases, $h$ has a local minimum or maximum in $(a,b)$.
 
-EOP
+QED
 
 #### Theorem 5.10 Mean Value Theorem
 

pages/Math4121/Math4121_L4.md

@@ -32,7 +32,7 @@ Hence, $g$ attains its infimum on $[a,b]$ at some $x\in (a,b)$. Then this $x$ is
 
 So $g'(x)=0$ and $f'(x)=\lambda$.
 
-EOP
+QED
 
 ### L'Hôpital's Rule
 
@@ -136,4 +136,4 @@ $$
 
 $\forall x\in (a,c_2)$, $\frac{f(x)}{g(x)}<q$.
 
-EOP
+QED

pages/Math4121/Math4121_L5.md

@@ -34,7 +34,7 @@ Set $F(x)=-f(x)$. and $q=-A+\epsilon>-A$. Apply main step, $\exists c_2\in (a,b)
 
 We take $c=\min(c_1,c_2)$. Then $\forall x\in (a,c)$, $\frac{f(x)}{g(x)}<q$.
 
-EOP
+QED
 
 ### Higher Order Derivatives
 
@@ -114,6 +114,6 @@ By Mean Value Theorem, $\exists x_n\in (\alpha,x_{n-1})$ such that $g^{(n)}(x_n)
 
 Since $g^{(n)}(\alpha)=0$ for $k=0,1,2,\cdots,n-1$, we can find $x_n\in (\alpha,x_{n-1})$ such that $g^{(n)}(x_n)=0$.
 
-EOP
+QED
 
 ## Chapter 6: Riemann-Stieltjes Integral

pages/Math4121/Math4121_L6.md

@@ -95,7 +95,7 @@ $$
 
 Same for $U(P_k,f,\alpha)\geq U(P_{k-1},f,\alpha)$.
 
-EOP
+QED
 
 #### Theorem 6.5
 
@@ -119,4 +119,4 @@ $$
 \underline{\int_a^b}f(x)d\alpha\leq \sup_{P_1}L(P_1,f,\alpha)\leq \inf_{P_2}U(P_2,f,\alpha)=\overline{\int_a^b}f(x)d\alpha
 $$
 
-EOP
+QED

pages/Math4121/Math4121_L7.md

@@ -50,7 +50,7 @@ $$
 
 So $f$ is Riemann integrable with respect to $\alpha$ on $[a, b]$.
 
-EOP
+QED
 
 #### Theorem 6.8
 
@@ -93,4 +93,4 @@ $$
 
 So, $f$ is Riemann integrable with respect to $\alpha$ on $[a, b]$.
 
-EOP
+QED

pages/Math4121/Math4121_L8.md

@@ -40,7 +40,7 @@ $$
 
 Therefore, $U(P,f,\alpha) - L(P,f,\alpha)<U(P,\alpha,f) - L(P,\alpha,f) < \epsilon$, so $f\in \mathscr{R}(\alpha)$ on $[a, b]$.
 
-EOP
+QED
 
 #### Theorem 6.10
 
@@ -76,7 +76,7 @@ Since $\epsilon$ is arbitrary, we have $U(P,f,\alpha) - L(P,f,\alpha) < \epsilon
 
 Therefore, $f\in \mathscr{R}(\alpha)$ on $[a, b]$.
 
-EOP
+QED
 
 #### Theorem 6.11
 

pages/Math4121/Math4121_L9.md

@@ -53,7 +53,7 @@ $$
 
 Since $\epsilon$ is arbitrary, $h\in \mathscr{R}(\alpha)$ on $[a, b]$.
 
-EOP
+QED
 
 ### Properties of Integrable Functions
 
@@ -108,4 +108,4 @@ So $U(P,h,\alpha)\leq U(P,f,\alpha)+U(P,g,\alpha)\leq \int_a^b f d\alpha + \int_
 
 Since $\epsilon$ is arbitrary, $\int_a^b h d\alpha \leq \int_a^b f d\alpha + \int_a^b g d\alpha$.
 
-EOP
+QED