The natural numbers are defined as either the positive integers (1, 2, 3, ...) or the nonnegative integers (0, 1, 2, ...). In either case there is a smallest one (either 0 or 1). And so if you use a notation that excludes the lower bound then to include the smallest natural number (either 0 or 1) you have to define your sequence in terms of a number outside the set, which is awkward.

A more practical example is that you are using unsigned integers, so the smallest number is 0. If you use an exclusive lower bound, you can't actually represent any range containing 0, because -1 is not a valid number you can store.