Prove the following bounds on European put options P(S, t), with expiry at time T, on an underlying share price S, with no dividends:

(a) P ≤ Ee^−r(T−t),

(b) P ≥ Ee^−r(T−t) − S,

(c) 0 ≤ P₂ − P₁ ≤ (E₂ − E₁)e^−r(T−t),

where P₁ and P₂ are calls with exercise prices E₁ and E₂ respectively, and E₁ < E₂.