ropy where we used the result in Eq. (B1). We can indefinitely repeat this operation of breaking down each of the single events into two other possibilities, and finally obtain the general rule: (2") = m A (2). Consider next A(S) = representing the information contained in S equiprobable events. It is clear from the previous demonstration that we also have the property: 1(Sm)=mA(S), meaning that the gain of information obtained by splitting the S initial events m times is precisely m. Since the result in Eq. (B4) applies for any integer S, we also have, for any integer T # S and n # m, 4(T") = n A(T) Step 2 With a given choice of n (sufficiently large), it is possible to find m for which we have the double inequality To convince ourselves of the validity of this result, we take the logarithm of both inequalities to obtain the following condition on the existence of m (given n, S, T): u-1<m<u, with = n log T/ log S. If n is large enough, we have U > 0, regardless of the values of S and T. According to Eq. (B7); the only two possibilities for m are (i) U is an integer: m=u, (ii) u is not an integer: m = E(u), where E(x) means the integer part of real x (e.g., E(2.75) = 2). The first case is straightforward. The second case is demonstrated by first setting U = E(u) + x and u - 1 = E(u - 1) + x where X is a real number satisfying 0 x < 1. Substituting these definitions in Eq. (B7) yields 0 X <m - E(u - -1)<1+x. Since m - E(u - 1) is nonzero, we have m > E(u - 1). Since m - E(u-1) is an integer; we also have m - E(u - 1) < 1. The only integer solution is m = E(u), since E(u) - E(u - 1) = 1. The general solution of Eq. (B6) is, thus, m = E(u) = E(n log log S). Appendix B Having shown that the property in Eq. (B6) is always valid for n sufficiently large, we perform the following operations: gS<nlogT<(m+1)log S Since n can be chosen arbitrarily large, the last result means that with E being made arbitrarily small (E = 1/n We now use the property that the function A(N) is monotonically increasing, as per requirement (2).