I'm kind of new to bioinformatics and trying to self-study. I'm reading a bioinformatics book: An Introduction to Bioinformatics Algorithms and ran into some problems about understanding the proof of the Breakpoint Reversal Sort algorithm in chapter 5.4.
Specifically, the book states a theorem:
If a permutation $\pi$ contains a decreasing strip, then there is a reversal $\rho$ that decreases the number of breakpoints in $\pi$, that is, breakpoint ($\rho(\pi)$) < breakpoint($\pi$).
This theorem guarantees that the algorithm will terminate and lead to no breakpoint. The book later explains the proof for the theorem:
Let $k$ be the smallest element of decreasing strip, so element $k-1$ would be the ending of an increasing strip. Therefore element $k$ and $k-1$ corresponds to a breakpoint. Then reversing the segment between $k$ and $k-1$ will decrease the number of breakpoints.
I'm quite confused about this proof and don't really understand what the authors mean by reversing the segment between $k$ and $k-1$. Can someone please explain the proof for me?
Here's the lecture slide about reversal sorting that based on the book http://csbio.unc.edu/mcmillan/Media/Lecture07Spring2015.pdf.