# Real analysis – Intuition behind the integrability of Riemann and partitions sequences

I began to study the theory of measurements and, as a review, the book I am using (Rana) begins to explain some theorems about the Riemann integrals and I try to fully understand the intuition and proofs of the following theorem:

Leave $$f$$ be an integrable function of Riemann in $$[a,b]$$, then there is a sequence of partitions. $$( pi_n) _n$$ such that $$pi_n subset pi_ {n + 1}$$ for all $$n geq 1$$.

I'm not even sure how to start, besides realizing that f is Riemann integrable $$int_ {a} ^ {b} f (x) dx = sup {L ( pi, f) } = inf {U ( pi, f) }$$ where $$pi$$ it is a partition of $$[a,b]$$. Then we can choose $$( pi_ {n} ^ {1}) _ {n}, ( pi_ {n} ^ {2}) n {}$$ such that $$lim$$ $$U ( pi_ {n} ^ {1}, f)$$=$$lim$$ $$L ( pi_ {n} ^ {2}, f) = int_ {a} ^ {b} f (x) dx$$ how n tends to $$infty$$.

After that I'm stuck. Any help would be really appreciated.