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.