Let $N=\{1, \ldots, n\}$ be the set of nodes of a network, where 1 is the source and $n$ is the $\operatorname{sink}$. Let $c_{i j}$ denote the capacity of the arc from node $i$ to node $j$.

(i) In the context of maximising the flow through this network, define the following terms: feasible flow, flow value, cut, cut capacity.

(ii) State and prove the max-flow min-cut theorem for network flows.