Hospital/Differentiable in inner interval/Fact/Proof

From Wikiversity
Jump to navigation Jump to search

Because has no zero in the interval and holds, it follows, because of fact, that is the only zero of . Let denote a sequence in , converging to .

For every there exists, due to fact, applied to the interval or , a (in the interior[1] of ,) fulfilling

The sequence converges also to , so that, because of the condition, the right-hand side converges to . Therefore, also the left-hand side converges to , and, because of , this means that converges to .

  1. The interior of a real interval

    is the interval without the boundaries.