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In a finite flow network, a maximum flow exists because the feasible flow set contains the zero flow and is a compact set: the capacity constraints bound every coordinate, and they and flow conservation define a closed set. The strength of a flow is a continuous function, so the extreme value theorem gives an attained maximum. An augmenting path in the residual network would permit positive augmentation, contradicting maximality. The source-reachable vertices in the residual network therefore define a cut of a flow network whose outgoing original edges are saturated and incoming original edges carry zero flow. Flow conservation makes the strength of a flow equal to this cut capacity, proving the max-flow min-cut theorem.

Existence and strong duality for finite real flow network edge capacities do not imply finite termination for every choice of augmenting paths in the Ford-Fulkerson algorithm. With irrational capacities, some choices can continue forever. With integer capacities, Integrality of the Ford-Fulkerson algorithm instead forces finite termination; rational capacities reduce to that case by a common denominator. Choosing shortest augmenting paths as in the Edmonds–Karp algorithm avoids dependence on capacity magnitudes.^[ref]

If every directed edge capacity increases by $\epsilon$, a cut of a flow network with $m(S)$ outgoing edges gains $\epsilon m(S)$. By the max-flow min-cut theorem, the new maximum flow value isFor a finite flow network, this is a nondecreasing concave piecewise-linear function, and its active cut can change as $\epsilon$ varies. Two unit-capacity edge-disjoint two-edge source-to-sink paths gain $2\epsilon$, refuting an invariably exact $\epsilon$ increase. If there is a directed source-to-sink path, every cut has at least one outgoing edge and the increase is at least $\epsilon$; without a path it can be zero.