Floating rate annuity

(a) A derivative contract pays αLT[T, T + α] at time T + α. By constructing a portfolio of ZCBs and a libor deposit that replicates the payout, prove that the value at t ≤ T of the derivative contract is Z(t, T) – Z(t, T + α).
(b) Let T0, T1, … , Tn be a sequence of times, with Ti+1 = Ti + α for a constant α > 0. Use your result from (a) to show that a floating leg of libor payments αLTi [Ti, Ti + α] at times Ti+1, i = 0, 1, … , n – 1, has value at time t ≤ T0 equal to a simple linear combination of ZCB prices.
(c) Hence find the value of a spot-starting infinite stream of libor payments, that is, when t = T0 = 0 and as n → ∞.