实分析| Real Analysis代写 MT3502

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这是一份andrews圣安德鲁斯大学 MT3502作业代写的成功案例

实分析| Real Analysis代写 MT3502
问题 1.

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
$$
F(b)-0=H(b)-0
$$


证明 .

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


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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 $9.6 .3$,
$$
\begin{aligned}
&F(d)=F(c)+\int_{c}^{-d} f, \
&G(d)=G(c)+\int^{-d} g
\end{aligned}
$$