The answer to your question may be somewhere in this thread.
The definition of the definite integral as the limit of an infinite sum, is just that, a definition. So it's true by definition. And yes, you have to accept it.
But I suspect that what you really meant to ask is whether this has been proven: Given a function, continuous on a closed interval [a,b], the difference between the values of the anti-derivative at a and b is the same as the definite integral of the function on the interval [a,b]. This is the fundamental theorem of calculus, and yes, like all theorems, it's been proven. If you didn't see the proof when you first learned integral calculus, either you were asleep or your teacher should be fired!