The recursion theoretic limit lemma, saying that each function with a $\varSigma_{n+2}$ graph is a limit of certain function with a $\varDelta_{n+1}$ graph, is provable in $\text{\rm B}\Sigma_{n+1}$.
@article{119409, author = {V\'\i t\v ezslav \v Svejdar}, title = {The limit lemma in fragments of arithmetic}, journal = {Commentationes Mathematicae Universitatis Carolinae}, volume = {44}, year = {2003}, pages = {565-568}, zbl = {1098.03067}, mrnumber = {2025821}, language = {en}, url = {http://dml.mathdoc.fr/item/119409} }
Švejdar, Vítězslav. The limit lemma in fragments of arithmetic. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) pp. 565-568. http://gdmltest.u-ga.fr/item/119409/
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