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Decide if the following definitions of Riemann integrability are equivalent.

Definition 1:

We say that $f:[a,b]\to \Bbb R$ is integrable if there exists an $L$ such that for any sequence of tagged paritions $P_n$ of $[a,b]$ satisfying $\|P_n\|\to 0$, we have that for given $\epsilon > 0$ there exists $n_0$ such that $n\ge n_0$ implies $$|S_P(f)-L|<\epsilon$$

Definition 2:

We say that $f:[a,b]\to \Bbb R$ is integrable if there exists an $L$ such that for all $\epsilon > 0$ there exists a tagged partition $P$ such that for any refinement $P*$ of $P$ we have that $$|S_{P*}(f)-L|<\epsilon$$

If such an $L$ exists, we call it $\int _a^b f$ in both cases.

I was given this an exercise, I think the definitions are equivalent, it is easy to see that $1$ implies $2$, but the other implication seems very hard to solve. Is anyone familiar with a proof of this?

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YoTengoUnLCD
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  • The Wiki article https://en.wikipedia.org/wiki/Riemann_integral (currently) does this by making a detour through the Darboux integral (not very elegantly however). – B. S. Thomson Feb 27 '16 at 01:41
  • Let $P_1$ be a partition of $[a,b]$ such that it has one partition point $p_1$ which is a partition point of $P$. Let $P_2$ be a refinement of $P_1$ such that $P_2$ has $p_1$ and $p_2$ as partition points, where $p_2$ is also a partition point of $P$. In general, let $P_n$ have $n$ of the partition points from P. Then any $n\geq n_0$ (where $n_0$ is the number of partition points of $P$, we have that $|S_P(f)-L|<\epsilon$. We now have a sequence of partitions such that $||P_n||\to 0$. We now have one sequence, but we need it to hold for any sequence. – smingerson Feb 27 '16 at 17:07
  • Not entirely sure that is something we can show easily. I have also only seen it shown using upper and lower sums. – smingerson Feb 27 '16 at 17:07
  • see http://math.stackexchange.com/a/1834341/72031 – Paramanand Singh Sep 09 '16 at 20:36

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