Summary and Section 5 — approximation in policy space reaches the minimal times, the unique solution of (3.2), after at most N − 1 iterations

Let $N = n + 1 \ge 2$ cities be given, with travel times $t_{ij} > 0$ for $i \ne j$. Let $f^{(k)}$ be the successive approximations (5.1) started from the direct-route policy (5.2) ($f_i^{(0)} = t_{iN}$ for $i \ne N$, $f_N^{(0)} = 0$). Then for every $k \ge N - 1$:

1. for every city $i$, $f_i^{(k)}$ is the minimal time (3.1) to travel from $i$ to $N$: it is the time of some route from $i$ to $N$, and no route from $i$ to $N$ is faster;
2. $f^{(k)}$ solves the system (3.2), $f_i^{(k)} = \min_{j\ne i}[t_{ij} + f_j^{(k)}]$ for $i \ne N$ and $f_N^{(k)} = 0$;
3. every real solution $F$ of (3.2) equals $f^{(k)}$.

In the paper's words, the algorithm "converges after at most $(N - 1)$ iterations" (Summary) to the solution of (3.2) (Section 5). In particular, the limit (5.6), $\lim_{k\to\infty} f_i^{(k)} = f_i$, holds and furnishes the solution of (3.2).

Formalization Note "Converges after at most $(N-1)$ iterations" is stated as equality for every $k \ge N - 1$, which is $k \ge n$ with $N = n + 1$. It is not stated as a limit. The bound is the paper's $N - 1$. $f_N^{(0)} = 0$ is the corrected reading of (5.2); see the (5.4) item. The minimal times are defined from routes, not from (3.2) or from the iteration.