Integrality and finite termination of the Ford–Fulkerson algorithm

Fix a network of $m$ directed arcs on nodes $\{0,\dots,n-1\}$, capacities $u_k\in[0,\infty]$, and distinct nodes $s\ne t$, with: no self-loops; every capacity strictly positive ($0<u_k$); every capacity either $\top=\infty$ or equal to the image of a natural number (so integer capacities are nonnegative integers); and an initial flow $f_0$ that is a feasible max-flow (conservation at nodes other than $s,t$; $(Nf_0)_t=-(Nf_0)_s$; $0\le f_{0,k}$ and $\mathrm{ofReal}(f_{0,k})\le u_k$) all of whose coordinates are integers. Conclusion, a conjunction: (a) every flow $f$ reachable from $f_0$ by zero or more Ford–Fulkerson steps (the reflexive–transitive closure of the step relation: current flow feasible, an augmenting path exists with finite bottleneck $\delta\ne\top$, and the new flow is $f+\delta_{\mathbb{R}}\chi$ along the path's $\pm1$ membership indicator $\chi$) has all coordinates integer; (b) if the max-flow value $\sup\{(Nf)_s : f\ \text{feasible}\}\ne+\infty$ (supremum in the extended reals), then there is no infinite sequence $g_0,g_1,g_2,\dots$ of flows with $g_0=f_0$ and every consecutive pair a Ford–Fulkerson step — i.e. the procedure cannot run forever. Nothing is claimed about which augmenting paths are chosen, and (b) is conditional on finiteness of the sup.