Solutions to open problems in Neeb's recent survey on   infinite-dimensional Lie groups
  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). 
Helge Glockner