Acceleration as an optimal polynomial

For $f(x)=\tfrac12x^\top Ax - b^\top x$ with $\mu I\preceq A\preceq LI$, show that any first-order method whose iterates lie in $x_0 + \operatorname{span}\{r_0,Ar_0,\dots,A^{t-1}r_0\}$ satisfies $x_t-x^\star = p_t(A)(x_0-x^\star)$ for some degree-$t$ polynomial with $p_t(0)=1$. Deduce that the best achievable worst-case rate is

$$\min_{p_t}\max_{\lambda\in[\mu,L]}|p_t(\lambda)| = \frac{1}{T_t\big(\frac{L+\mu}{L-\mu}\big)} \approx 2\left(\frac{\sqrt\kappa-1}{\sqrt\kappa+1}\right)^t,$$

where $T_t$ is the Chebyshev polynomial of the first kind.