Geodesic
A sufficient condition for the recursive convergence of a monotone sequence
Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part IV, Tome 20 (1971), pp. 97-103 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is shown that a bounded monotone recursive sequence of recursive real numbers is recursively convergent provided its members satisfy a recurrent equation of some (rather general) form. 
@article{ZNSL_1971_20_a9,
     author = {Yu. V. Matiyasevich},
     title = {A~sufficient condition for the recursive convergence of a~monotone sequence},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {97--103},
     year = {1971},
     volume = {20},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1971_20_a9/}
}
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Yu. V. Matiyasevich. A sufficient condition for the recursive convergence of a monotone sequence. Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part IV, Tome 20 (1971), pp. 97-103. http://geodesic.mathdoc.fr/item/ZNSL_1971_20_a9/