(where \(\{t\}\) denotes the fractional part of \(t\text{,}\) and similarly for \(\{x\},\{y\}\)).
Proof.
Observe that \(\lfloor t \rfloor\) is a step function with jump \(1\) at each integer, so
\begin{equation}
\sum_{y \lt n \leq x} f(n) = \int_y^x f d\lfloor t \rfloor\text{.}\tag{4.6.2}
\end{equation}
We apply linearity of the Riemann-Stieltjes integral with respect to \(t = \lfloor t \rfloor + \{t\}\text{,}\) so that \(dt = d\lfloor t \rfloor + d\{t\}\text{.}\) This gives
\begin{equation}
\sum_{y \lt n \leq x} f(n) = \int_y^x f dt - \int_y^x f d\{t\}\text{.}\tag{4.6.3}
\end{equation}
