Let {a_n} be a sequence of real numbers defined by

    a₀ ∉ {0,1},
    a₁ = 1 - a₀, and
    a_n+1 = 1 - a_n(1 - a_n) for all n ≥ 1.

Let P_n and S_n be defined by

    P_n = a₀a₁a₂ ··· a_n and

    S_n = 1/a₀ + 1/a₁ + 1/a₂ + ··· + 1/a_n

for all n ≥ 0.

Prove the following

    P_nS_n = 1 for all n ≥ 0.