# Spivak Calculus vs Abbott

I have a copy of both Spivak calculus and Apostol.

I use both from time. However, I find some of Spivak's problems just boring. I was thinking of maybe trying Abbott.

Especially for continuity proofs, I want a book which slowly, progresses from easy to difficult proofs, because Spivak is very fast, and very difficult. Most of the time I cant do Spivak problems.

So should I give Abbott's Understanding Analysis, a try? Or is Spivak better?

• Spivak is very fast?! He takes a detailed, intuitive step-by-step approach to define the notion of limit and hence of continuity. He takes seven pages to do so, and the first four of them contain pictures and diagrams. – Pedro Tamaroff Dec 31 '14 at 9:52
• @PedroTamaroff, I phrased it incorrectly. Some of the examples he shows of proofs are very easy compared to the exercises. The exercises are really really difficult. – Amad27 Dec 31 '14 at 10:05
• Could you give an example? – Pedro Tamaroff Dec 31 '14 at 10:10
• If you were able to show that $f(x)\leqslant g(x)\implies \lim\limits_{x\to a} f(x)\leqslant \lim\limits_{x\to a} g(x)$, consider $f\leqslant g\leqslant h$. If you let $a=\lim\limits_{x\to a} f=\lim\limits_{x\to a} h,b=\lim\limits_{x\to a} g,$, the last consideration shows that $\lim\limits_{x\to a} f = a\leqslant b =\lim\limits_{x\to a} g$, $\lim\limits_{x\to a} g =b\leqslant a = \lim\limits_{x\to a} h$, so the double inequality entails $a=b$. – Pedro Tamaroff Dec 31 '14 at 10:22
• Many exercises are like this: one carries into another, in increasing (or not?) difficulty. Personally, I would keep reading Spivak, and give it more time if necessary. You're always invited to drop by the chat and clear out any doubts or request help. – Pedro Tamaroff Dec 31 '14 at 10:23