  Let $\Gamma$ be a cofinite Fuchsian subgroup. The canonical Green's function
associated with $\Gamma$ arises in Arakelov theory when establishing
asymptotics for Arakelov invariants of the modular curve associated with some
congruence subgroup of level $N$ with a positive integer $N$. More precisely,
in the known cases, canonical Green's functions at certain cusps contribute to
the analytic part of the asymptotics for the self-intersection of the relative
dualizing sheaf. In this article, we prove canonical Green's function of a
cofinite Fuchsian subgroup at cusps bounded by the scattering constants, the
Kronecker limit functions, and the Selberg zeta function of the group $\Gamma$.
Then as an application, we prove an asymptotic expression of the canonical
Green's function associated with $\Gamma_0(N)$, for any positive integer $N$.
