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In the book of Mathematical Analysis by Zorich, at page 114, it is given that

if the partition $P'$ of the interval $I$ is obtained by refining intervals of the partition $P$, then $$s(f,P) \leq s(f,P') \leq S(f,P') \leq S(f,P),$$ where $s$ and $S$ denotes the lower and upper Darboux sums of $f$.

I can see the result intuitively, but having hard time to put it into a formal language, so how can we prove this result formally ?

I mean any help or hint about the general sketch of the proof are also welcomed.

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Suppose you have an interval $[a,b]$ and that $c\in(a,b)$. Then\begin{align}\min f|_{[a,b]}.(b-a)&=\min f|_{[a,b]}.(b-c)+\min f|_{[a,b]}.(c-a)\\&\leqslant\min f|_{[c,b]}.(b-c)+\min f|_{[a,c]}.(c-a)\end{align}and, of course$$\max f|_{[a,b]}.(b-a)\geqslant\max f|_{[c,b]}.(b-c)+\max f|_{[a,c]}.(c-a).$$This is why when you refine the partion the upper sums get lower and the lower sums get bigger.

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  • $\begingroup$ Thanks a lot for your answer, this was, actually, my intuitive explanation for the theorem also. $\endgroup$ – onurcanbektas Jan 31 '18 at 11:46
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    $\begingroup$ Use this to prove the statement when $P'=P\cup\{c\}$ for some $c$ and then use induction. $\endgroup$ – José Carlos Santos Jan 31 '18 at 12:05

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