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Proving existence of certain step functions implies integrability

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In problem 26(b) of Chapter 13 of Spivak's Calculus, the following problem is given

Suppose that for all $\epsilon>0$, there are step functions $s_1\le f$ and $s_2\ge f$ such that $\int_a^bs_2 -\int_a^bs_1<\epsilon$. Prove that $f$ is integrable.

The proposed solution by the author is

The existence of such step functions implies the existence of partitions $P_1$ and $P_2$ with $U(f,P_1)-L(f,P_2)<\epsilon$.

For context, the definitions on integrability and step-functions in the text are as follows:

integrability: A bounded function $f$ is said to be integrable on $[a,b$ iff for every $\epsilon>0$, there exists a partition $P$ of $[a,b]$ such that $U(f,P)-L(f,P)<\epsilon$

Step function: A function defined on $[a,b]$ is called a step function if there is a partition $P=\{t_0 ,t_1,...,t_n\}$ such that $s$ is constant on each $(t_{k-1},t_k)$ (the values of s on $t_k$ may vary)

I think Spivak's proposed solution suggests using the fact that $$\int_a^bs\le L(f,P_1)\le U(f,P_2)\le\int_a^b t $$

But I don't think that the first and last inequalities are necessarily true. It is easy to show that the integrals of the step functions are given by $$\int_a ^b s=\sum_{k=1}^ns_k(x_k-x_{k-1})\qquad \text{and}\qquad \int_a ^b t=\sum_{k=1}^n t_k(x_k-x_{k-1})$$where $s_k$ and $t_k$ denote the constant values of $s$ and $t$ on $(x_{k-1},x_k)$ respectively. If the supremum and infimum of $f$ on $[x_{k-1},x_k]$ occur in the interior, then they lie between $s_k$ and $t_k$ and we have the proposed inequality. But what about the case where the supremum/infimum occur at the endpoints and are smaller/larger than $t_k$ / $s_k$?


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