An $s$-stage explicit Runge-Kutta method of order $p$, with constant step size $h>0$, is applied to the differential equation $y^{\prime}=\lambda y, t \geqslant 0$.

(a) Prove that

$$y_{n+1}=P_{s}(\lambda h) y_{n}$$

where $P_{s}$ is a polynomial of degree $s$.

(b) Prove that the order $p$ of any $s$-stage explicit Runge-Kutta method satisfies the inequality $p \leqslant s$ and, for $p=s$, write down an explicit expression for $P_{s}$.

(c) Prove that no explicit Runge-Kutta method can be A-stable.