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I've started working through Spivak's Calculus. I'm going into senior year after this summer, took the AP Calculus BC test last year, and wanted to get a firmer foundation in calculus before I take Calculus III next year. I'm still struggling through Chapter 1, unfortunately. I understand the reading material, but there are quite a lot of problems to do. A lot of them are repetitive and not very difficult, but very long, so I found a syllabus online and I'm doing those as well as all of the ones marked with asterisks. I have no prior experience writing proofs though, so I'd like to verify that I'm doing them acceptably. The first one turned out terrible convoluted, and I'm sure it could have been done with much more brevity. I'm still not quite sure what I'm allowed to assume other than the postulates given. I'm not sure how I should link it, is Google Drive okay?

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You wish to show that $$0 < a < b \Longrightarrow a < \sqrt{ab} < \frac{a+b}{b} < b$$

Since $0 < a < b$, we can multiply by $a$ and get $0 < a^2 < ab$. Then we take the principal square root, giving $ 0 < a < \sqrt{ab}$. This shows the first inequality.

Now, expanding $(\sqrt{a} - \sqrt{b})^2 > 0$, it gives $a + b - 2\sqrt{ab} > 0$, which in turn gives $(a+b)/2 > \sqrt{ab}$. This shows the second inequality.

Now, again considering $0 < a < b$, we multiply by $1/2$, giving $ 0 < a/2 < b/2$. Now, we add $b/2$. This gives $b/2 < \frac{a+b}{2} < b$. This shows the third inequality.

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