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.