Having trouble with a problem set related to partitions and integrals, I have a somewhat understanding of the subject as I've solved problems related to it already, but these are rather hard. I think I just need a starting nudge, so to speak (studying independently so no prof. to ask for that 
)
Let f be a real integrable function on [a,b] and P=[a=x[SUB]0[/SUB], x1... xn=b)] a partition. Let si, ti be two arbitrary points on [xi-1, xi]. Show that
I:
∑(i=1,n)|f(si)-f(ti)|(xi-xi-1) < ϵ
II:
|∑(i=1,n)f(ti)(xi-xi-1)|-(∫(a,b)f dx) < ϵ
III:
Show that if there exists a differentiable function F on [a,b] so that F'=f, then
∫(a,b)f(x) dx=F(b)-F(a)
Thanks!
)Let f be a real integrable function on [a,b] and P=[a=x[SUB]0[/SUB], x1... xn=b)] a partition. Let si, ti be two arbitrary points on [xi-1, xi]. Show that
I:
∑(i=1,n)|f(si)-f(ti)|(xi-xi-1) < ϵ
II:
|∑(i=1,n)f(ti)(xi-xi-1)|-(∫(a,b)f dx) < ϵ
III:
Show that if there exists a differentiable function F on [a,b] so that F'=f, then
∫(a,b)f(x) dx=F(b)-F(a)
Thanks!