This paper proves an area law for the entanglement entropy in one-dimensional gapped quantum systems. The entanglement entropy of a region scales with the boundary area, not the volume. The bound on the entropy grows exponentially with the correlation length, which is surprising. The paper discusses quantum expanders and conjectures that completely positive maps may provide an alternative way to derive the area law. It also shows that the bound on von Neumann entropy implies a bound on Rényi entropy for sufficiently large α < 1 and that the ground state can be approximated by a matrix product state. The paper explores the reasons for believing that the entanglement entropy of a gapped system obeys an area law. It notes that in one dimension, conformal field theory calculations show that the entanglement entropy is bounded and diverges proportionally to the correlation length near a critical point. In higher dimensions, systems represented by matrix product states obey an area law. However, there is no general proof of an area law, despite the exponential decay of correlation functions in gapped systems. The existence of data hiding states and quantum expanders shows that large entanglement can exist even with small correlations, complicating the proof of an area law. The paper provides a proof of an area law for one-dimensional systems under the assumption of a gap. The result bounds the entanglement entropy by a quantity that grows exponentially in the correlation length, which is much faster than the linear growth expected. The paper also derives bounds for gapped local systems that inter-relate the von Neumann entropy, the Rényi entropy, and the error in approximating the ground state by a matrix product state. The paper defines the lattice and Hamiltonian, and introduces notation for the ground state and reduced density matrix. It then presents a theorem and proof of the area law, using the Lieb-Robinson bound and properties of matrix product states. The proof involves showing that the entropy of a region cannot exceed a certain bound, leading to a contradiction if the bound is violated. The paper also discusses the implications of the result for numerical simulations of one-dimensional quantum systems and conjectures that completely positive maps may provide an alternative way to prove an area law.This paper proves an area law for the entanglement entropy in one-dimensional gapped quantum systems. The entanglement entropy of a region scales with the boundary area, not the volume. The bound on the entropy grows exponentially with the correlation length, which is surprising. The paper discusses quantum expanders and conjectures that completely positive maps may provide an alternative way to derive the area law. It also shows that the bound on von Neumann entropy implies a bound on Rényi entropy for sufficiently large α < 1 and that the ground state can be approximated by a matrix product state. The paper explores the reasons for believing that the entanglement entropy of a gapped system obeys an area law. It notes that in one dimension, conformal field theory calculations show that the entanglement entropy is bounded and diverges proportionally to the correlation length near a critical point. In higher dimensions, systems represented by matrix product states obey an area law. However, there is no general proof of an area law, despite the exponential decay of correlation functions in gapped systems. The existence of data hiding states and quantum expanders shows that large entanglement can exist even with small correlations, complicating the proof of an area law. The paper provides a proof of an area law for one-dimensional systems under the assumption of a gap. The result bounds the entanglement entropy by a quantity that grows exponentially in the correlation length, which is much faster than the linear growth expected. The paper also derives bounds for gapped local systems that inter-relate the von Neumann entropy, the Rényi entropy, and the error in approximating the ground state by a matrix product state. The paper defines the lattice and Hamiltonian, and introduces notation for the ground state and reduced density matrix. It then presents a theorem and proof of the area law, using the Lieb-Robinson bound and properties of matrix product states. The proof involves showing that the entropy of a region cannot exceed a certain bound, leading to a contradiction if the bound is violated. The paper also discusses the implications of the result for numerical simulations of one-dimensional quantum systems and conjectures that completely positive maps may provide an alternative way to prove an area law.