From Bolzano‐Weierstraß to Arzelà‐Ascoli

Abstract

Abstract. 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 [9] 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 fn : [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 BolzanoWeierstraß 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. We will give two different formalizations of this theorem, show how these can be reduced to suitable instances of the Bolzano-Weierstraß principle and, using this, obtain a classification of them in the sense of reverse mathematics and computable analysis. Bolzano-WeierstraßIn [9] we investigated the strength of the following two variants of the BolzanoWeierstraß principle:• The (strong) Bolzano-Weierstraß principle (BW) is the statement that each bounded sequence of real numbers contains a subsequence converging at the rate 2 −k . (This is the usual formulation in reverse mathematics. The rate 2 −k stems from the fact that real numbers are coded as sequences that converge at this rate. However, 2 −k is just an arbitrarily chosen rate. In fact, one can easily convert a sequence converging at a given rate into a sequence converging at any other given rate.)2010 Mathematics Subject Classification. Primary 03F60; Secondary 03D80, 03B30.

Bounded variation and the strength of Helly's selection theorem

“…This continues our work in [10] and [12] where (instances of) the Bolzano-Weierstraß principle and the Arzelà-Ascoli theorem were analyzed. There we showed, among others, that an instance of the Arzelà-Ascoli theorem is equivalent to a suitable single instance of the Bolzano-Weierstraß principle (for the unit interval [0, 1]), which, in turn, is equivalent to an instance of WKL for Σ 0 1 -trees.…”

“…In [12,Lemma 3,Corollary 4] we showed that an equicontinuous sequence of functions h n : [0, 1] −→ [0, 1] converges uniformly iff h n converges pointwise on Q ∩ [0, 1], i.e., for an enumeration q of Q ∩ [0, 1] the sequence (h n (q(i))) i n ⊆ [0, 1] N converges in [0, 1] N with the product norm d((x i ), (y i )) = i 2 −i d(x i , y i ). Moreover, from a rate of convergence and the modulus of uniform equicontinuity one can calculate a rate of convergence of h n in • ∞ .…”

“…With this the Arzelà-Ascoli theorem follows directly from an application of the Bolzano-Weierstraß principle for the space [0, 1] N . For details see [12].…”

“…This proposition is provable in ACA 0 since Bolzano-Weierstraß principle for the space [0, 1] N is instance-wise equivalent to the Bolzano-Weierstraß principle for [0, 1] which is provable in ACA 0 , see e.g. [12,18].…”

“…This corollary should be compared with Theorem 3.1 of [10], where it is shown that BW is instance-wise equivalent to WKL for Σ 0 1 -trees, and Theorem 9 of [12], where it is shown that BW is instance-wise equivalent to the Arzelà-Ascoli theorem.…”