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It says that the Riemann Integral on[a,b] exists, if for any epsilon this holds:

|Riemannsum(f(x) on [a,b] )- Integralvalue|< epsilon, for any partion over [a,b] where the norm of the partion is less than delta.

Now comes my problem, lets say I want to prove that the Riemann-Integral of x on[a,b] is

(b^2-a^2)/2.

I would then choose a partion (b-a)/n, and show that for a given epsilon, I can make n as big as I want, so that every Riemann sum of this partion goes to the desired value. I would do this by first making the Riemann sum as small as possible with a given n, by choosing the left value at each subinterval, and then as big as I can for a given n, by choosing the right value at each subinterval.

Now comes my question, does this really prove that x is Riemann-integrable over [a,b], the problem is when we look at the definiton. It says that it is supposed to hold for every partion where the norm goes to zero. But I have only proved it for a partion where each subinterval as equal length (b-a)/n. Isn't this a really big problem with the definition. I mean you can choose that the norm of the partion goes to zero with very many partions, how does one solve this problem?