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Let $f$ be bounded on $[a,b]$ and integrable on $[c,b]$ for $a<c<b$. I need to prove the following are equivalent:

a) $\lim\limits_{x\to a+}\int_{x}^{b}f$ exists in $\mathbb{R}$

b)$\lim\limits_{n\to \infty}\int_{a_n}^{b}f$ exists in $\mathbb{R}$ for a monotonically decreasing sequence $a_n\to a$

c) $f$ is integrable on $[a,b]$.

I know I need to prove $a\implies b$, $b\implies c$, and $c\implies a$, but I don't even know where to get started. Definitions I have are patitions, darboux sums/ integrals and reimann integrals. I do not have improper integrals, the fundamental thm of calculus, nor integral MVT.

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  • $\begingroup$ You can prove things in the order you suggest but this is not necessary. For example you can show equivalence of (a) and (c) (consider the partition $a,c,x_2,x_3,\dots b $ ($c$ generic or as needed, number between $a,b $). This can be taken as a generic partition or particular, for particular chiloice of $c $. Now you can try and show that for every small $\epsilon>0$, there a partition such that the difference of upper and lower sums is smaller than this (check precise statement of theorem). This should be possible using (a) and the hypothesis. $\endgroup$ – AnyAD Nov 20 '18 at 3:34
  • $\begingroup$ Well a) $\implies $ b) is obvious from definition of limit. And a) also implies that $f$ is Riemann integrable on $[c, b] $ for all $c\in(a, b) $. This along with boundedness of $f$ implies c). Same argument applies for b) $\implies $ c). Further c) $\implies $ a) is obvious. $\endgroup$ – Paramanand Singh Nov 20 '18 at 13:38

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