# 实分析| Real Analysis代写 MT3502

and $c_{n} \rightarrow a, d_{n} \rightarrow b$; then
$$F\left(d_{n}\right)-F\left(c_{n}\right)=H\left(d_{n}\right)-H\left(c_{n}\right)$$
for all $n$, so in the limit we have
$$F(b)-0=H(b)-0$$

by the continuity of $F$ and In other words,
$$\int_{a}^{b} f=\int_{a}^{b} f \cdot \diamond$$

## MT3502 COURSE NOTES ：

Let $F:[a, b] \rightarrow \mathbb{R}$ and $G:[a, b] \rightarrow \mathbb{R}$ be the indefinite upper integrals of $f$ and $g$, respectively:
$$F(x)=\int_{a}^{-x} f \text { and } G(x)=\int_{a}^{-x} g$$
for $a \leq x \leq b$. If $a<c<d<b$ then, as in the proof of $9.6 .3$,
\begin{aligned} &F(d)=F(c)+\int_{c}^{-d} f, \ &G(d)=G(c)+\int^{-d} g \end{aligned}