Consider the random walk on the integer number line $\mathbb{Z}$ which starts at 0 and at each step $k$ moves $\epsilon_k=+1$ or $\epsilon_{k}=-1$ with equal probability $1/2$. The probability that a return to $0$ occurs at time $2n$ is denoted by $u_{2n}$ and the chance that a first return to origin occurs at $t=2n$ is denoted by $f_{2n}$.
Let $s_{2n}$ denote the cumulative sum at time $2n$,
In Feller's book, An Introduction to Probability Theory and Its Applications, in section 4 on page 75, the author goes on the prove that the probability of the first return to origin at time $2n$ is
$\begin{align} P(s_1\ne 0, s_2 \ne 0, \ldots, s_{2n-1}\ne 0, s_{2n} \ne 0) &= 1 - f_{2} - f_{4} - \ldots- f_{2n}\\ &= 1 - (1 - u_{2}) - (u_{2} - u_{4}) - (u_{4} - u_{6}) - \ldots - (u_{2n-2}-u_{2n})\\ &= u_{2n} \end{align}$
It is not clear to me, how he equates the probabilities of the events $f_{2}=1-u_2$, $f_{4}=u_{2}-u_{4}$, $\ldots$ etcetera.
It would be really helpful, if someone could elaborate this fine point.
Cheers!