From Bolzano-Weierstraß to Arzelà-Ascoli

We show how one can obtain solutions to the Arzelà-Ascoli theorem using suitable applications of the Bolzano-Weierstraß principle. With this, we can apply the results from \cite{aK} and obtain a classification of the strength of instances of the Arzelà-Ascoli theorem and a variant of it. Let AA be the statement that each equicontinuous sequence of functions f_n: [0,1] --> [0,1] contains a subsequence that converges uniformly with the rate 2^-k and let AA_weak be the statement that each such sequence contains a subsequence which converges uniformly but possibly without any rate. We show that AA is instance-wise equivalent over RCA_0 to the Bolzano-Weierstraß principle BW and that AA_weak is instance-wise equivalent over WKL_0 to BW_weak, and thus to the strong cohesive principle StCOH. Moreover, we show that over RCA_0 the principles AA_weak, BW_weak + WKL and StCOH + WKL are equivalent.

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