  A remarkable feature of typical ground states of strongly-correlated
many-body systems is that the entanglement entropy is not an extensive
quantity. In one dimension, there exists a proof that a finite correlation
length sets a constant upper-bound on the entanglement entropy, called the area
law. However, the known bound exists only in a hypothetical limit, rendering
its physical relevance highly questionable. In this paper, we give a simple
proof of the area law for entanglement entropy in one dimension under the
condition of exponentially decaying correlations. Our proof dramatically
reduces the previously known bound on the entanglement entropy, bringing it,
for the first time, into a realistic regime. The proof is composed of several
simple and straightforward steps based on elementary quantum information tools.
We discuss the underlying physical picture, based on a renormalization-like
construction underpinning the proof, which transforms the entanglement entropy
of a continuous region into a sum of mutual informations in different length
scales and the entanglement entropy at the boundary.
