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 .
- ↑
The
interior
of a
real interval
is the interval without the boundaries.