  We show how one can obtain solutions to the Arzel\`a-Ascoli theorem using
suitable applications of the Bolzano-Weierstra{\ss} principle. With this, we
can apply the results from \cite{aK} and obtain a classification of the
strength of instances of the Arzel\`a-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-Weierstra{\ss} 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.
