I am reading the paper above by Lee, Gharan and Trevisan, and I am having trouble with lemma 4.8 on page 23. How do we form the set $T_1, \ldots, T_{r'}$ such that $r'>(1-\delta/2)k$ and each set $T_i$ satisfies both the upper bound and lower bound with respect to $\epsilon$ as in the paper ?I am struggling with this for a while, so any help is greatly appreciated.
Multi-way spectral partitioning and higher-order Cheeger inequalities
spectral graph theory