Let $f,g : [a,b] \to R$ be bounded and suppose that $f(x)$ not equal to $g(x)$ only at a finite number of points.
Prove that the lower Riemann integrals of $f$ and $g$ are equal to each other, and prove the same case with the upper integrals.
I have been told to use the fact that $L(f,P)$ tends to the lower integral given that the mesh of partition tends to $0$ (same for upper integral).
I've been thinking of expanding $L(f,P) - L(g,P)$ and commenting on how the infimums will be equal for many intervals and showing that that tends to $0$ as the mesh of $P$ tends to $0$ and therefore the two limits equal each other, implying the lower Riemann integrals are equal.
Need some help with the details however!