What is the definition of S in the proof for convergence of Riemann sum?

[Werg22]

In a book of mine, the author proceeds to the proof that a Riemann sum in a interval [a,b] must converge by proving that for S_m and S_n (m>n) where the span of the subdivisions is suffiencienly small, then

|S_m - S_n)| < e(b-a)

Where e can assume infinitly small values in dependence of the span.

Now I understand why S_m has to be bounded, however I do not see an argument strong enough for convergeance - couldn't S_m assume constantly changing lower or higher values within a certain interval? That certainly could satisfy the inequality above... What am I missing?

 

[mathwonk]

you need some hypothesis on the function f, like monotonicity or continuity, or boundedness and oiecewise continuity, or somehing, since bad behavior is surely possible.

 

[matt grime]

What are S_m and S_n? There is no such thing as the canonical subdivision S_n and S_m.
And what do you mean by 'in dependance'? If indeed you have shown that given e>0, there is an N such that for m>n>N then |S_m-S_n|<e(b-a) then you have indeed shown existence of the integral, since S_m is then a Cauchy sequence.

 

[Werg22]

What are S_m and S_n? There is no such thing as the canonical subdivision S_n and S_m.
And what do you mean by 'in dependance'? If indeed you have shown that given e>0, there is an N such that for m>n>N then |S_m-S_n|<e(b-a) then you have indeed shown existence of the integral, since S_m is then a Cauchy sequence.

m and n refer to the number of subdivisions. The value "e" is equivalent to MAX [|f(x+p) - f(x)|] where p is the length of the span of S. I know this is a Cauchy sequence since the inequality shows that S is bounded... my question lies in the uncertainty of the convergance of S, as it could not converge towards a specific value but still satisfy the condition.

 

[matt grime]

That does not specify S_n at all. There are uncountably many partitions into n subdivisions. A cauchy sequence converges in the reals, by the way, so there is no problem here at all, if indeed the sequence is cauchy.

And what is S? This is a new piece of terminology.

 

[shmoe]

What is S_n though? If you won't tell us how your book defined it, there's not much we can do (there's no universal way to define these things as matt has mentioned).

What is S? Is it your sequence of S_n? It seems like it might be, and that you think it is a Cauchy sequence. If so, there should be no problem, a Cauchy sequence converges to something.

You need to give more details on the definitions and other assumptions like conditions on f. Or provide the reference, someone might have it handy and be willing to help.

edit- beaten to the punch, but you should see a pattern emerging here. We'll keep asking for specifics until you supply them.