I am studying the chapter of markov processes on Durrett's PTE and get stuck with one step in one of the examples (5.5.14 in the fifth edition), in which he argues about the positive recurrence of a M/G/1 queue when $\mu<1$. We denote in the following discussion, the stopping time $$ T_x:=\inf\{n>0:X_n=x\}. $$
In that example he first shows that $E_xT_0\leq\frac{x}{1-\mu}$ and he then claimed that the inequality is actually an equality. In that step he used the equation $E_xT_{x-1}=E_1T_0$ for $x\geq1$. This seems very intuitive but I am not sure how to rigorously argue for it using the markov property.
Below is the excerpt which contains the argument that I am concerned about.
