This is an excerpt from Calculus Answers in regards to showing that the inequality $M_i' - m_i \leq M_i - m_i$ (where $M_i, m_i$ define the $\inf$ and $\sup$ of any interval when computing Riemann sums).
I'm uncertain how he came to the equality of $M_i'$ and $M_i$; wouldn't the inequality $-m_i \leq M_i$ mean $M_i' \leq M_i$? Can anyone shed some light on this?