Books that provide justifications of perturbations and asymptotic methods. I am looking for books which provide justifications (proofs of appropriate theorems ) of various perturbation methods.
In particular I would like to study about justification of matched asymptotic expansions, multiply scales,
WKB, Poincare method. 
Thank you in advance. 

I would say that a nice source to base your search on is
C. Kuehn, Multiple Time Scale Dynamics, Springer (2015), ISBN 978-3-319-12315-8, [link].
The book is very extensive concerning perturbation methods. In addition, it not only contains proofs, but also an extensive literature section, which guides you to literature where necessary proofs can be found.
