  There are several notions of convergence for sequences of bounded degree
graphs. One such notion is left convergence, which is based on counting
neighborhood distributions. Another notion is right convergence, based on
counting homomorphisms to a target (weighted) graph. Borgs, Chayes, Kahn and
Lov\'asz showed that a sequence of bounded degree graphs is left convergent if
and only if it is right convergent for certain target graphs $H$ with all
weights (including loops) close to $1$. We give a short alternative proof of
this statement. In particular, for each bounded degree graph $G$ we associate
functions $f_{G,k}$ for every positive integer $k$, and we show that left
convergence of a sequence of graphs is equivalent to the convergence of the
partial derivatives of each of these functions at the origin, while right
convergence is equivalent to pointwise convergence. Using the bound on the
maximum degree of the graphs, we can uniformly bound the partial derivatives at
the origin, and show that the Taylor series converges uniformly on a domain
independent of the graph, which implies the equivalence.
