terrytosh said:
Or is reading a proofs book enough
Linear Algebra, at least for me, was easier than analysis.
Depending on the analysis book you choose to study from. It may require a bit of background from linear algebra. Ie., what is an n-tuple , norm of a vector , and some other topics.
Or the analysis book can be independent of linear algebra. Ie., it is a review of single-variable calculus, but now you work through the theory of it.
A very good intro analysis book is Understanding Analysis by Abbot.
A good Linear Algebra (theory) is Linear Algebra Done Right by Axler.
You do not necessarily need to read a proofs book, but it does not hurt to read one. Especially if you are not familiar proofs.
A good proof method book is by Hammok. A pdf of the book can be legally downloaded.