  Local convergence techniques have become a key methodology to study sparse
random graphs. However, convergence of many random graph properties does not
directly follow from local convergence. A notable, and important, such random
graph property 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 give several
examples where this method gives rise to a novel law of large numbers for the
giant, based on results proved in the literature. Aside from these examples, we
apply our novel method to the configuration model as a proof of concept,
reproving a well-established result. As a side result of this proof, we
identify the small-world nature of the configuration model.
