Update Math4121_L9.md

pages/Math4121/Math4121_L9.md

@@ -63,7 +63,7 @@ Let $f,g\in \mathscr{R}(\alpha)$ on $[a, b]$.
 
 (a) $f+g\in \mathscr{R}(\alpha)$ on $[a, b]$, $\int_a^b (f+g)d\alpha = \int_a^b f d\alpha + \int_a^b g d\alpha$. (Linearity of the integral)
 
-If $c\in \mathbb{R}$, then $cf\in \mathscr{R}(\alpha)$ on $[a, b]$, and $\int_a^b cf d\alpha = c\int_a^b f d\alpha$.
+(aa) If $c\in \mathbb{R}$, then $cf\in \mathscr{R}(\alpha)$ on $[a, b]$, and $\int_a^b cf d\alpha = c\int_a^b f d\alpha$.
 
 (b) If $f(x)\leq g(x),\forall x\in [a, b]$, then $\int_a^b f d\alpha \leq \int_a^b g d\alpha$.
 
@@ -73,6 +73,39 @@ If $c\in \mathbb{R}$, then $cf\in \mathscr{R}(\alpha)$ on $[a, b]$, and $\int_a^
 
 (e) If $f\in \mathscr{R}(\beta)$ then $f\in \mathscr{R}(\alpha+\beta)$ and $\int_a^b f d(\alpha+\beta) = \int_a^b f d\alpha + \int_a^b f d\beta$.
 
+Proof:
 
+Property (aa), (b), (e) holds for Riemann Sums themselves.
 
+$$
+\sup cf(x) = c\sup f(x)\quad \forall c\in \mathbb{R}
+$$
 
+$$
+U(P,cf, \alpha) = cU(P,f,\alpha)
+$$
+
+For (b), notice that if $f(x)\leq g(x)$, then $\sup f(x)\leq \sup g(x)$, $U(P,f,\alpha)\leq U(P,g,\alpha)$. and $L(P,f,\alpha)\leq L(P,g,\alpha)$.
+
+For (e), notice that
+
+$$
+\begin{aligned}
+\Delta (\alpha+\beta)_i &= \alpha(x_i)-\alpha(x_{i-1})+\beta(x_i)-\beta(x_{i-1}) \\
+&= \Delta \alpha_i + \Delta \beta_i
+\end{aligned}
+$$
+
+(c),(d) are left as homework.
+
+For (a), Set $h(x)=f(x)+g(x)$. Then $h\in \mathscr{R}(\alpha)$ on $[a, b]$ and we will show $\int_a^b h d\alpha \leq \int_a^b f d\alpha + \int_a^b g d\alpha$.
+
+Since $f,g\in \mathscr{R}(\alpha)$ on $[a, b]$, for any $\epsilon > 0$, there exists a partition $P_1,P_2$ of $[a, b]$ such that $U(f,P_1,\alpha)-L(f,P_1,\alpha) < \epsilon$ and $U(g,P_2,\alpha)-L(g,P_2,\alpha) < \epsilon$.
+
+Let $P=P_1\cup P_2$. Then $U(P,f,\alpha)\leq U(P_1,f,\alpha)< \int_a^b f d\alpha + \epsilon$ and $U(P,g,\alpha)\leq U(P_2,g,\alpha)< \int_a^b g d\alpha + \epsilon$.
+
+So $U(P,h,\alpha)\leq U(P,f,\alpha)+U(P,g,\alpha)\leq \int_a^b f d\alpha + \int_a^b g d\alpha + 2\epsilon$.
+
+Since $\epsilon$ is arbitrary, $\int_a^b h d\alpha \leq \int_a^b f d\alpha + \int_a^b g d\alpha$.
+
+EOP