 Solution

For a strictly decreasing function, $f(n+1)<f(x)<f(n)$ in the interior of $[n,n+1]$. Integrating over the unit interval and summing from $n=r$ to $s-1$ gives the integral comparison for a nonincreasing nonnegative function:Geometrically, right-endpoint rectangles lie below the curve and left-endpoint rectangles above it. If “decreasing” permits flat pieces, use non-strict inequalities unless strictness is ensured on some interval. When $r=1$, the endpoint value $f(1)$ is also needed; the power function below is defined there.