From the textbook:
Let $P=\left\{I_1,I_2,\ldots,I_n\right\},{\ }Q=\left\{J_1,J_2,\ldots,J_m\right\}$ be partitions of $[a,b]$, where $Q$ is a refinement of $P$, so $m\geq n$. Since $Q$ is a refinement of $P$, each interval $I_k$ in $P$ is an almost disjoint union of intervals in $Q$, which we can write as $$I_k=\bigcup_{\ell=p_k}^{q_k} J_\ell$$ for some indices $p_k\le q_k$. If $p_k<q_k$, then $I_k$ is split into two or more smaller intervals in $Q$, and if $p_k=q_k$, then $I_k$ belongs to both $P$ and $Q$. Since the intervals are listed in order, we have $$p_1=1,{\ }p_{k+1}=q_k+1,{\ }q_n=m$$
If, for example, given an interval [0,1], a passable partition $P$ and refinement of $Q$ as $$ P=\left\{\left[0,\frac{1}{5}\right],\left[\frac{1}{5},1\right]\right\} $$ $$ Q =\left\{\left[0,\frac{1}{5}\right],\left[\frac{1}{5},\frac{1}{4}\right],\left[\frac{1}{4},\frac{1}{3}\right],\left[\frac{1}{3},\frac{1}{2}\right],\left[\frac{1}{2},1\right]\right\} $$
For $k=1$ we have the interval $\left[0,\frac{1}{5}\right]$, then how $$I_{k=1}=\bigcup_{l=p_1}^{q_1}J_l$$ should work?
Thanks.