calculus - Proof that monotone functions are integrable with the classical definition of the Riemann Integral

Let $f:[a,b]\to \mathbb{R}$ be a monotone function (say strictly increasing).

Then, do for every $\epsilon>0$ exist two step functions $h,g$ so that $g\le f\le h$ and $0\le h-g\le \epsilon$?

Does there exist some closed form of these functions like the one below?

I encountered the problem when trying to prove the Riemann Integrability of monotone functions via the traditional definition of the Riemann Integral (not the one with Darboux sums). The book I am reading proves that a continuous function is integrable via the process below:

Let $\mathcal{P}=\left\{ a=x_0<... be a partition of $[a,b]$. Define $m_i=\min_{x\in [x_{i-1},x_i]}f(x)$ and $M_i=\max_{x\in [x_{i-1},x_i]}f(x)$. By the Extreme Value theorem $M_i,m_i$ are well defined. We approximate $f$ with step functions:

$$\begin{gather}g=m_1\chi_{[x_0,x_1]}+\sum_{i=2}^nm_i\chi_{(x_{i-1},x_i]}\\ h=M_1\chi_{[x_0,x_1]}+\sum_{i=2}^nM_i\chi_{(x_{i-1},x_i]}\end{gather}$$

It is easily seen that they satisfy the "step function approximation" and as $0\le \int_a^bh-g\le \epsilon$ (uniform continuity) by a previous theorem $f$ is integrable.

The book then goes on to generalise by discussing regulated functions. I would like however to see a self contained proof similar to the previous one if $f$ is monotone. This question is reduced to the two questions asked in the beggining of the post

Answer

Let $\varepsilon>0$ and $N_\varepsilon$ the smallest $n \in \mathbb{N}$ such that

$$\frac{1}{n}(b-a)(f(b)-f(a)) \le \varepsilon$$
For $n \ge \max\{2,N_\varepsilon\}$ consider the following partition of $[a,b]$:
$$

\mathcal{P}=\{a=x_0 $$Set$$
A_i= $$\begin{cases} [x_0,x_1] & \text{ for } i=0\\ (x_i,x_{i+1}]& \text{ for } 1 \le i \le n-1 \end{cases}$$
$$Since $f$ is strictly increasing for each $i \in \{0,\ldots,n-1\}$ we have$$
f(x_i) \le f(x) \le f(x_{i+1}) \quad \forall\ x_i \le x \le x_{i+1}.
$$Setting$$
h=\sum_{i=0}^{n-1}f(x_{i+1})\chi_{A_i},\ g=\sum_{i=0}^{n-1}f(x_i)\chi_{A_i}
$$we have$$
g(x)\le f(x) \le h(x) \quad \forall\ x \in [a,b],

$$and$$
h(x)-g(x)=\sum_{i=0}^{n-1}\Big(f(x_{i+1})-f(x_i)\Big)\chi_{A_i}(x)>0 \quad \forall\ x \in [a,b].
$$
In addition
$$\begin{eqnarray} \int_a^b(h-g)&=&\sum_{i=0}^{n-1}(f(x_{i+1})-f(x_i))\int_a^b\chi_{A_i}=\sum_{i=0}^{n-1}(f(x_{i+1})-f(x_i))(x_{i+1}-x_i)\\ &=&\frac{b-a}{n}\sum_{i=0}^{n-1}(f(x_{i+1})-f(x_i))=\frac{1}{n}(b-a)(f(b)-f(a))\\ &\le&\frac{1}{N_\varepsilon}(b-a)(f(b)-f(a))\le \varepsilon. \end{eqnarray}$$