Consider the Fibonacci sequence f₁, f₂,..., where f₁ = 1, f₂ = 1, and

fn = fn - 1 + fn - 2
for n ≥ 3. Then
f₃ = f₁ + f₂ = 1 + 1 = 2,
f₄ = f₂ + 3 = 1 + 2 = 3,

Please show by mathematical induction that every integer n ≥ 1 can be expressed as the sum of distinct Fibonacci numbers, no two of which are consecutive.