  Local convergence techniques have become a key methodology to study random
graphs in sparse settings where the average degree remains bounded. However,
many random graph properties do not directly converge when the random graph
converges locally. A notable, and important, random graph property that does
not follow from local convergence is the size and uniqueness of the giant
component. We provide a simple criterion that guarantees that local convergence
of a random graph implies the convergence of the proportion of vertices in the
maximal connected component. We further show that, when this condition holds,
the local properties of the giant are also described by the local limit.
  We apply this novel method to the configuration model as a proof of concept,
reproving a result that is well-established. As a side result this proof, we
show that the proof also implies the small-world nature of the configuration
model.
