Integral bounds for a finite harmonic sum ($\ln ((N+1)/(m+1))<{\sum }_{n=m+1}^{N}1/n<\ln (N/m)$)

For integers $N>m\ge 1$, compare the decreasing function $1/x$ with unit rectangles: on $n-1<x<n$, $1/(x+1)<1/n<1/x$. Integrating and summing over $n=m+1,\dots ,N$ proves the displayed strict bounds. Adding a known harmonic number $H_m$ gives bounds on $H_N$ without computing every omitted term. A requested rounding is certified when both bounds round to the same value; a midpoint estimate alone does not establish the rounding.