Prove that the theoretical value of a swap is independent

Question

Swap frequency:

We have assumed that payment dates for the fixed and floating legs of a swap contract are the same. However, in practice, the frequency may differ-for example, in the US swap market the fixed leg is usually semi-annual (α = 0.5), versus a floating leg of threemonth libor paid quarterly (α = 0.25).

(a) Prove that the theoretical value of a swap is independent of the floating frequency-for example, three-month libor paid quarterly has the same value as six-month libor paid semi-annually.

(b) Prove that the value of a swap with fixed rate K is not independent of the fixed leg frequency. In particular, does P_t[T₀, T_n] increase or decrease if α changes from 0.25 to 1? Deduce directly whether the annual swap rate for T₀ to T_n is higher or lower than the quarterly swap rate for the same period. Assume the same daycount convention.