Hospital/Differentiable in inner interval/Fact/Proof
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Proof
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 .
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The interior of a real interval
is the interval without the boundaries.