Approximation of classical theorems by constructive ones is considered. (By constructive proofs the authors understand those proofs that do not use the Excluded Middle principle.) The paper develops some ideas of G. Kreisel, P. S. Novikov and T. Coquand. To develop their notion of approximation the authors consider a representation of statements by trees equipped with Brouwer's topology. Quoting the authors, ``we provide a family of statements, each corresponding to an effective construction, and whose limit, with respect to Brouwer's topology, is (the tree form of) \(A\).'' The paper contains a series of examples and some philosophical remarks. The following references (added to the references in the paper) would probably help to see the problem of approximation in a larger context: \textit{G. E. Mints}, ``Transfinite expansions of arithmetic formulas'' [J. Sov. Math. 10, 533-547 (1978); transl. from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 49, 51-66 (1975; Zbl 0319.02026)]; \textit{M. G. Gel'fond}, ``Relationship between the classical and constructive developments of mathematical analysis'' [J. Sov. Math. 6, 347-352 (1976); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 32, 5-11 (1973; Zbl 0354.02028)]; \textit{N. A. Shanin}, ``On a hierarchy of methods of understanding of sentences in constructive mathematics'' (Russian) [Tr. Mat. Inst. Steklova 129, 203-266 (1973; Zbl 0312.02028)].