Question

Define riemann integrability in two different ways and prove their equivalence?

Answer 1

We say that the Riemann integral of f equals s if the following condition holds:

For a given ε>0, there exists δ such that for any tagged partition x0,⋯,xn and t0,⋯,tn−1 whose mesh is less than δ, we have

∣∣∣∑i=0n−1f(ti)(xi+1−xi)−s∣∣∣<ε.

Unfortunately, this definition is very difficult to use. It would help to develop an equivalent definition of the Riemann integral which is easier to work with. We develop this definition now, with a proof of equivalence following.(?) Our new definition says that the Riemann integral of f equals s if the following condition holds:

For all ε>0, there exists a tagged partition x0,⋯,xn and t0,⋯,tn−1 such that for any refinement y0,⋯,ym and s0,⋯,sm−1 of x0,⋯,xn and t0,⋯,tn−1, we have

∣∣∣∑i=0m−1f(si)(yi+1−yi)−s∣∣∣<ε.

Hope this helps you ☺️☺️✌️✌️❤️❤️