I have been taking a university course in which they define the Riemann-Stieltjes Integral using upper sums and lower sums of a function and then we take their respective infimum and supremum.
In the course they have also mentioned Reimann sums and they have proved that if the limit of reimann sums, as the mesh of the tagged partitions go to zero, exists then it is Riemann-Stieltjes Integrable. But they said that the converse holds true when the function if continuous or the function you integrate if with respect to is continuous.
I also found this pdf https://www.math.mcgill.ca/labute/courses/255w03/L1.pdf Riemann-Stieltjes using the limit of reimann sums.
But according to our professor the two definitions are not strictly equivalent. So which definition is correct?