  We solve three open problems concerning infinite-dimensional Lie groups posed
in a recent survey article by K.-H. Neeb:
  (1) There exists a subgroup of some infinite-dimensional Lie group G which
does not admit an initial Lie subgroup structure;
  (2) The pathology cannot occur if G is a direct limit of an ascending
sequence of finite-dimensional Lie groups;
  (3) Every such direct limit group is a ``topological group with Lie algebra''
in the sense of Hofmann and Morris.
  Moreover, we prove a version of Borel's Theorem announced in the survey,
ensuring the existence of compactly supported smooth diffeomorphisms with given
Taylor series around a fixed point p (provided the tangent map at p has
positive determinant).
