A book with simple proof exercises and full answers

Question

I am quite bad at basic proof strategies, and in order to get some practice at this, I'm working through Daniel Solow's How to Read and Do Proofs (third ed.) Unfortunately, I'm having the problem I have with every single book I've tried to go through on proofs, which is that I'll hit an exercise that I can't overcome, and which there is no answer for in the back of the book. It's 2am, and I've spent more time than I really want to think about on an exercise which was clearly not meant to cause someone this much distress.

So firstly, I'd very much appreciate it if someone could help me with this exercise:

suppose you have already proved the proposition that "If a and b are nonnegative real numbers, then $(a + b)/2 \ge \sqrt{ab}$

Explain how you could use this proposition to prove that if a and b are real numbers satisfying the property that $b \ge 2|a|$, then $b \ge \sqrt{b^2 - 4a^2}$. Be careful how you match up notation.

This is almost certainly an exercise in making sure I don't get my as and bs mixed up, but I cannot for the life of me figure out how I'm actually supposed to approach it. It seems qualitatively harder than all the other neighbouring exercises, and doesn't seem to have any correlate in the preceding chapter of the book.

Secondly, does anyone have a better recommendation for a book on basic proof techniques with all the sodding answers in the back, so I don't lose even more sleep in a frustrated and futile attempt to complete an exercise I have no hope of figuring out?

Answer 1

$\sqrt {b^2 - 4a^2} = \sqrt{(b - 2a)(b + 2a)} $ due to the given conditions both of these factors are positive numbers. I suggest you check that yourself. Therefore, a direct application of your result yields,

$$\sqrt {b^2 - 4a^2} = \sqrt{(b - 2a)(b + 2a)} \le \frac {(b - 2a) + (b + 2a)} {2} = b$$

Now, about this sleepless problem. This is my humble recommendation. What you need is not a book with solutions in the back. I'm thinking you are still in your early stages and the sleepless nights you spend worrying over a problem are extremely valuable.

If you have the hunger and the will, don't take the easy route out. Keep persevering because that will eventually help you overcome your problems and become more potent in mathematics. Reading solutions off the book will never help you get there.

Something my old teacher used to say: "There are no shortcuts to learn mathematics".

Answer 2

if $b\geq2|a|$ then $b^2\geq4a^2$ so $b^2-4a^2\leq b^2$

Using this we know

$\sqrt{b^2-4a^2}\leq\sqrt{b^2}=b$ which was to be proven.

Edit: Regarding sleeplesness problem: try to kay off math for at least an hour before going to sleep, it helps you, also try to do excercise to be tired enough to sleep like a baby.