I am reading the appendix to chapter 13 in Spivak's Calculus (third edition, Cambridge University Press) and am stuck on the proof of theorem 1.
Theorem 1: Suppose that $f$ is integrable on $[a,b]$. Then for every $\epsilon>0$ there is some $\delta>0$ such that, if $P=\{t_0,\ldots,t_n\}$ is any partition of $[a,b]$ with all lengths $t_i-t_{i-1}<\delta$, then $$\left|\sum_{i=1}^n f(x_i)(t_i-t_{i-1})-\int_a^b f(x)\mathrm{d}x\right|<\epsilon,$$for any Riemann sum formed by choosing $x_i$ in $[t_{i-1},t_i]$.
The subsequent proof has two parts; when $f$ is continuous and when it is not. I will skip the first part of the proof, since I understand this part (see here for the first part of the proof).
Proof: The argument in the general case is simple (though perhaps a bit messy), using Problem 13-27, which says that there are continuous functions $g\leq f\leq h$ satisfying $$\int_a^b g \leq \int_a^b f \leq \int_a^b h \tag{1},$$ with $$\int_a^b h-\int_a^b g<\epsilon.$$ We have $$\sum_{i=1}^n g(x_i)(t_i-t_{i-1})\leq \sum_{i=1}^n f(x_i)(t_i-t_{i-1})\leq \sum_{i=1}^n h(x_i)(t_i-t_{i-1}),$$ and since the theorem holds for continuous functions, we know that for $t_i-t_{i-1}<\delta$, the left- and right-hand sides of this inequality are close to the left- and right-hand sides of $(1)$. This implies that the two middle terms $$\int_a^b f \ \text{and} \ \sum_{i=1}^n f(x_i)(t_i-t_{i-1}),$$ must be close to $\int_a^b h-\int_a^b f$, which is small. Detailed inequalities are left to the skeptical reader.
First of all, I suspect $\int_a^b h-\int_a^b f$ is a typo. He means probably $\int_a^b h-\int_a^b g$. Secondly, I am trying to work out the detailed inequalities which Spivak omitted, but am stuck on this part. The goal is to obtain $$\underbrace{\left|\sum_{i=1}^n f(x_i)(t_i-t_{i-1})-\int_a^b f\right|}_{(*)}\leq \ldots<\epsilon.$$ I have tried finding an upper bound to $(*)$ and add in $0$, but without being able to make it all smaller than $\epsilon$. Grateful for any help!
