What is wrong with equation 7.3 in Sutton & Barto's book?

Question

Equation 7.3 of Sutton Barto book: $$\text{Equation: } max_s|\mathbb{E}_\pi[G_{t:t+n}|S_t = s] - v_\pi| \le \gamma^nmax_s|V_{t+n-1}(s) - v_\pi(s)| $$ $$\text{where }G_{t:t+n} = R_{t+1} + \gamma R_{t+2} + .....+\gamma^{n-1} R_{t+n} + \gamma^nV_{t+n-1}(S_{t+n})$$ Here $V_{t+n-1}(S_{t+n})$ is the estimate of $V_\pi(S_{t+n})$

But the Left Hand Side of the above equation should be zeros as, for any state s, $G_{t:t+n}$ is an unbiased estimate of $v_\pi(s)$ hence $\mathbb{E}_\pi[G_{t:t+n}|S_t = s] = v_\pi(s)$.

Answer 1

In general, $\mathbb{E}_\pi[G_{t:t+n}|S_t = s] \neq v_\pi(s)$. $v_\pi(s)$ is defined as $\mathbb{E}_\pi[\sum_{k=0}^{\infty} \gamma^k R_{t+k+1} | S_t = s]$, so you should be able to see why the two are not equal when the LHS is an expectation of the $n$th step return. They would only be equal as $n \rightarrow \infty$.