Proposition 2.1 -- gambler's ruin

For every natural number $n$ with $0 < n$ and every state $k$ in the finite set $\{0, 1, \dots, n\}$, the theorem asserts the conjunction of two identities about the "gambler's ruin" transition matrix $P$ on the $(n+1)$-element state space $\{0,\dots,n\}$, defined entrywise by: if the current state is $0$ or $n$, then $P(k,k)=1$ and all other entries in that row are $0$ (both endpoints are absorbing); otherwise (for $0 < k < n$) the entry $P(k,l)$ is $1/2$ when $l = k+1$ or $l = k-1$, and $0$ for every other $l$. First conjunct: the quantity "probability of hitting $n$ before $0$ starting from $k$", defined as the infinite series $\sum_{t=0}^{\infty} f_t$, where $f_t$ is the finite sum, over all sequences $\omega = (\omega_0, \omega_1, \dots, \omega_t)$ of states satisfying (i) $\omega_0 = k$, (ii) $\omega_t = n$, (iii) $\omega_i \neq n$ for every index $i \neq t$, and (iv) $\omega_i \neq 0$ for every index $i$, including $i = 0$, of the path weight $\prod_{i=0}^{t-1} P(\omega_i, \omega_{i+1})$ (an empty product, equal to $1$, when $t = 0$), equals the real number $k/n$ (real division; here $n > 0$, so this is ordinary division). Note that when $k = n$ the length-zero path contributes $1$ at $t = 0$, and when $k = 0$ condition (iv) excludes every path, so the series is identically $0$. Second conjunct: the quantity "expected hitting time of the set $\{0, n\}$ starting from $k$", defined as the infinite series $\sum_{t=0}^{\infty} q_t$, where $q_t$ is the finite sum, over all sequences $(\omega_0, \dots, \omega_t)$ of states satisfying $\omega_0 = k$ and $\omega_i \notin \{0, n\}$ for every index $i$, including $i = 0$, of the same path weight $\prod_{i=0}^{t-1} P(\omega_i, \omega_{i+1})$, equals the real number $k \cdot (n - k)$ (an ordinary product of reals; since $k \le n$ here, $n - k$ is the usual nonnegative difference). Note that when $k \in \{0, n\}$ every term of this series vanishes, so the left side is $0$. In both conjuncts the infinite series over $t$ is interpreted with the convention that a series which fails to converge (is not summable) is assigned the value $0$; when the series converges, the value is its ordinary sum.