I have been following Feller's proof of Basic Limit theorem (page 336 and following, An introduction to probability theory and its applications, W. Feller). It's basically the one that says that the probability of a being in a state of an ergodic chain converges to a stationary value in the long time limit and this value is $1/\mu$, the mean recurrence time. I am almost done. I got stuck with one of the final arguments, at page 338.
where $f_i$ are numbers such that $\sum_{i}f_i=1$ and $0\leq f_i\leq 1$. $\mu$ is defined as $\mu=\sum_i if_i$. Variables $u_i$ represent the probability of the stochastic event to verify at trial number $i$, while $f_i$ represent the probability of seeing the event for the first time after $i$ trials. As such, you have the relation called by the book (11.1): $u_n=u_{n-1}f_1+u_{n-2}f_2+...+u_0f_n$. But now if I sum over $n$ as the book says, I get something like: $$\sum_n^N u_n=(u_0f_1)+(u_1f_1+u_0f_2)+...+(u_{N-1}f_1+..+u_0f_n)=$$ $$u_0(1-\rho_N)+u_1(1-\rho_{N-1})+...+u_{N-1}(1-\rho_{1})$$. Bringing all terms with just $u_n$ to the side of the summation at the top left I am left with a $$u_N=u_N\rho_0=-(u_0\rho_N+u_1\rho_{N-1}+...+u_{N-1}\rho_{1})$$. What the book says it's one appears to be zero to me. It's a real pity, only this calculation and I am done understanding the theorem ! I give a link to the book if you need it: https://it.1lib.to/book/966574/a18835
Don't forget the limits of summation!
The identity obtained is $$\sum_{n=1}^N u_n=(u_0f_1)+(u_1f_1+u_0f_2)+...+(u_{N-1}f_1+..+u_0f_N)=$$ $$u_0(1-\rho_N)+u_1(1-\rho_{N-1})+...+u_{N-1}(1-\rho_{1}) \,.$$ Note that $u_0$ does not appear on the Left hand side. Since $u_0=1$, that explains the missing 1.
When encountering such contradictions, checking for small $N$ will usually clear things up.