  We consider the Braunstein-Kimble protocol for continuous variable
teleportation and its application for the simulation of bosonic channels. We
discuss the convergence properties of this protocol under various topologies
(strong, uniform, and bounded-uniform) clarifying some typical
misinterpretations in the literature. We then show that the teleportation
simulation of an arbitrary single-mode Gaussian channel is uniformly convergent
to the channel if and only if its noise matrix has full rank. The various forms
of convergence are then discussed within adaptive protocols, where the
simulation error must be propagated to the output of the protocol by means of a
"peeling" argument, following techniques from PLOB [arXiv:1510.08863]. Finally,
as an application of the peeling argument and the various topologies of
convergence, we provide complete rigorous proofs for recently-claimed strong
converse bounds for private communication over Gaussian channels.
