# Prove the following are equivalent

Let $$f$$ be bounded on $$[a,b]$$ and integrable on $$[c,b]$$ for $$a. 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.

• 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. – AnyAD Nov 20 '18 at 3:34
• 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. – Paramanand Singh Nov 20 '18 at 13:38