Analysis Texts Advice

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I am after a nice analysis textbook which predominately covers inequalities and asymptotics. Something with some number theory, such as a very elementary analytic number theory text would be great!

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It seems to me you'd be better off in the long run complementing a standard source like Amann and Escher (the first volume) or Rudin (the first eight chapters suffice, especially the eighth in your case; the key is doing the exercises), which are modern texts that emphasise topology, with Hardy's A Course of Pure Mathematics or Whittaker and Watson (which have more the flavour you seem to be after but is somewhat dated) or, more to your interests, something like- http://www.amazon.com/Asymptopia-Student-Mathematical-Library-Spencer/dp/1470409046/

Then you can try something more specialised. For analytic number theory there's the excellent text of Bordellès, or that of Apostol (both minimal in prerequisites). For asymptotics Dover offers a reprint of de Bruijn's Asymptotic Methods in Analysis.

I realise this may not be quite what you were after, but if this is for a first course in analysis it's better not to compromise modern foundations and use supplementary resources to suit your inclinations.