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I am having a lot of trouble with this question. I think that I am meant to set up 3 partitions, $P,\; \alpha\;\text{and} \; \psi$, where $P={\{x_0, x_1, \cdots,x_n\}}$, $\alpha={\{x_0, x_1, \cdots,x_k\}}$ and $\psi={\{x_k, x_{k+1}\cdots,x_n\}}$. Then I attempted to use the integrability criterion that for an arbitrary $\epsilon > 0$ we have

$$ U(f,P) - L(f,P) < \epsilon $$ and then I think I am meant to use that inequality with the fact that $U(f,\alpha)\leq U(f,P)$ and that $U(f,\psi)\leq U(f,P)$. But this is where I lose track. I have read some things also about the infinimum and supremum but I do not understand their applications in this question.

John Warts
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    If $P$ is a partition of $[a,b]$ and $P'$ is another partition that includes all of $P$ plus also the point $c$, then $L(f,P) \leq L(f,P') \leq U(f,P') \leq U(f, P)$. Furthermore, $P'$ can be broken up into two partions: one over $[a,c]$ and the other over $[c,b]$ – Ben Apr 28 '21 at 16:28
  • @Ben Ok, but how do I know the inequality for $P''$ and $P'''$? Because the new partitions from a to c and c to b are smaller than their containers, I am having trouble writing the inequality for it. – John Warts Apr 28 '21 at 17:13
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    Suppose $\exists P$ s.t. $U(f,P) - L(f,P) < \epsilon$. Then we can find a $P'$ that includes the point $c$ s.t. $U(f,P') - L(f,P') < \epsilon$. We can split $L(f,P')$ into $L(f,P_{ac})$ and $L(f,P_{cb})$, and similarly for the upper sum. You will be left with something like $U(f,P_{ac}) - L(f,P_{ac}) + U(f,P_{cb}) - L(f,P_{cb}) < \epsilon$. But you know by definition for any partition $U - L \geq 0$. This should let you conclude something about each U L pair... – Ben Apr 28 '21 at 17:33
  • @Ben thanks so much for your help mate. – John Warts Apr 28 '21 at 17:48
  • You're very welcome. So far we're close to showing that integrability on $[a,b]$ implies of integrability on $[a,c]$ and on $[c,b]$. We'll still need to show the reverse, and also that the integral on $[a,b]$ is equal to the sum of the other two. For this last part you will probably want to use supremum/infimum arguments. – Ben Apr 28 '21 at 18:14
  • @Ben that's alright, my textbook has some instructions that I found that worked ok. Thanks again. – John Warts Apr 28 '21 at 18:31

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