I have a question about finding out if a function is integrable and what is the integral. My question is about taking a partition which divides every subinterval to equal length.

I will take an easy example. f(x)=x^{2 }, [0,b] , where b is a positive real number. I have uploaded a picture of what I did.

Now It is clear to me why it is integrable, because we have created squence of partitions , and U(f,Pn)-L(f,Pn)-> 0 when n tends to infinity.

Which means for every epsilon there exists N such that for every n>N we get U(f,P)-L(f,P)<epsilon which means that it's integrable.

Now my question is , we defined two sets , I will call them with other names so it won't be confusing.

A={U(f,P) : P is the partition of the interval [0,b]}

B={L(f,P) : P is the partition of the interval [0,b]}

supA=infB then f is integrable at the given interval.

But when we take the limit when n tends to infinity , the supremum and infimum we get is not supA and infB because we took the set of all partitions with its subintervals having same length. Now don't I have to show that the supremum that I have found is actually supA? or the infimum that I have found is infB? And if so, how can I do that? Or can I create two sets we will call them C and D

C={U(f,P) : P is the partition of the interval [0,b], every subinterval has the same length}

D={L(f,P) : P is the partition of the interval [0,b], every subinterval has the same length}

Since we have supC<supA<infB<infD (the notation here is acctually less or equal)

But since we have found supC=infD we actually come to the conclusion supC=supA=infB=infD

Can I do it like this? And if this is true, is it redundant? and if it is why is it redundant? finding out that U(f,P)-L(f,P)<epsilon is enough to come to the conclusion that the inf and sup or two sets are equal? Thank you.