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Pyrrhus

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Spivak's book is by far my favourite. He conveys the subject elegantly yet rigorously, while maintaining his writing in a way that tells a definitive "story" of analysis. I find that mathematical texts fall into to general categories: Dictionary style, with theorem after theorem, and a very mechanical structure and tone to the writing (see Stephen Friedberg's "Linear Algebra"); or a less rigorous colloquial style, sort of like a high-school textbook where the material is "dumbed down" for supposed "ease of learning" (see Randall Knight's "Physics for Scientists and Engineers"). Spivak's book manages to avoid both of these categories. He also manages to subtly poke fun at several formalities that mathematicians like to uphold, which I thought was nice touch.

The problems are hard but very insightful. Keep in mind however, that to really learn mathematics you must internalize it, and no one exposition can do that for you. Even Spivak's book has some gray areas (particularly his treatment of sequences and series), so arm yourself with one good book to focus on, and several other minor resources that can help clarify ideas and build upon your intuition.

The problems are hard but very insightful. Keep in mind however, that to really learn mathematics you must internalize it, and no one exposition can do that for you. Even Spivak's book has some gray areas (particularly his treatment of sequences and series), so arm yourself with one good book to focus on, and several other minor resources that can help clarify ideas and build upon your intuition.

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