I started to self-study measure theory and as a review the book Im using(Rana) begin explaining some theorems on Riemann Integrals and im trying to understand to the fullest the intuition and proofs of the next theorem:
Let $f$ be a Riemann integrble function on $[a,b]$, then there exists a sequence of partitions $(\pi_n)_n$ such that $\pi_n\subset \pi_{n+1}$ for all $n\geq 1$.
Im not even sure how to start, besides noticing that since f is Riemann integrable $\int_{a}^{b}f(x)dx=sup\{L(\pi,f)\}=inf\{U(\pi,f)\}$ where $\pi$ is a partition of $[a,b]$. Then we can choose $(\pi_{n}^{1})_{n},(\pi_{n}^{2})_{n}$ such that $lim$ $U(\pi_{n}^{1},f)$=$lim$ $L(\pi_{n}^{2},f)=\int_{a}^{b}f(x)dx$ as n tends to $\infty$ and such that $lim$ $U(\pi_{n},f)=$$lim$ $L(\pi_{n},f)=\int_{a}^{b}f(x)dx$.
After that Im stuck. Any help would be really appreciated.