  We extract verified algorithms for exact real number computation from
constructive proofs. To this end we use a coinductive representation of reals
as streams of binary signed digits. The main objective of this paper is the
formalisation of a constructive proof that real numbers are closed with respect
to limits. All the proofs of the main theorem and the first application are
implemented in the Minlog proof system and the extracted terms are further
translated into Haskell. We compare two approaches. The first approach is a
direct proof. In the second approach we make use of the representation of reals
by a Cauchy-sequence of rationals. Utilizing translations between the two
represenation and using the completeness of the Cauchy-reals, the proof is very
short.
  In both cases we use Minlog's program extraction mechanism to automatically
extract a formally verified program that transforms a converging sequence of
reals, i.e. a sequence of streams of binary signed digits together with a
modulus of convergence, into the binary signed digit representation of its
limit. The correctness of the extracted terms follows directly from the
soundness theorem of program extraction.
  As a first application we use the extracted algorithms together with Heron's
method to construct an algorithm that computes square roots with respect to the
binary signed digit representation. In a second application we use the
convergence theorem to show that the signed digit representation of real
numbers is closed under multiplication.
