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

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 $$\mathbb{E}_\pi[G_{t:t+n}|S_t = s] = v_\pi(s)$$.

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 $$n \rightarrow \infty$$.