= Solution
Cutting an open-boundary <matrix product state> across one virtual bond of dimension $\chi$ gives <Schmidt rank> at most $\chi$, so its <entanglement entropy> obeys
$$
S(A)\leq\log\chi.
$$
A periodic interval cuts two bonds and obeys $S(A)\leq2\log\chi$. Since the boundary of a one-dimensional interval has a constant number of points, this is an area law.
For a <projected entangled pair state>, a bipartition crossing $n_\partial$ virtual bonds has Schmidt rank at most $\chi^{n_\partial}$. The <tensor-network area-law bound> is therefore
$$
\boxed{S(A)\leq n_\partial\log\chi}.
$$
Because $n_\partial$ is proportional to the lattice boundary area, every fixed-bond-dimension PEPS satisfies an area law.
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