Ideally, Casper would be a stepping stone to a purely Proof-of-Stake system, one in which we don't need farms of computers wasting energy to solve meaningless computational problems.
In Bolzano's case, the aim of providing a reconstruction of parts of analysis and geometry, so that the exposition would use only "explanatory" proofs, also led to major mathematical results, such as his purely analytic proof of the intermediate value theorem.
A purely algebraic proof of Livingston's theorem was given by Zimmermann [380], while a nice sketch of proof was given by Dunfield and Thurston [141].
He essentially formulated the modern criterion of convergence in his Purely Analytic Proof (Bolzano 1817, 11 f).
Effros and Ruan [3] showed that one can give a purely metric proof of this important theorem by using a technique of Pisier [4] and Haagerup [5].
Bolzano [1817] expressed this attitude with regard to the intermediate value theorem for the real numbers (IVT) before giving a purely analytic proof, arguing that spatial thinking could not be used to help justify the IVT.
In his Purely analytical proof (1817), too, Bolzano proves first the general theorem for two continuous functions (Russ 2004, § 15, 274) and only in § 18 (p. 276 7) for one continuous function.
The approach taken in it led to a better understanding of Riemann surfaces and, especially, to a purely algebraic proof of the celebrated Riemann-Roch Theorem (Geyer 1981, Kolmogorov & Yushkevich 2001, ch. 2).
Effros and Ruan [36] showed that one can give a purely metric proof of this important theorem by using the technique of Pisier [37] and Haagerup [38] (as modified in [39]).
Effros and Ruan [15] showed that one can give a purely metric proof of this important theorem by using a technique of Pisier [16] and Haagerup [17] (as modified in [18]).
In this paper, we give a purely functional analytic proof of this fact.
