Triangle Inequality and the Like

I didn’t finish a careful proof of the triangle inequality in class, so I am presenting a more polished argument here. I abandoned the text’s argument in favor of what I hope is a more intuitive (albeit longer) proof.

I’ve also had a number of requests for hints on a couple homework problems, specifically numbers 3.7 and 3.8. So here are my comments on those problems:

Problem 3.7 This problem asks us to prove that $$|b| 0$$, for otherwise the assumption $$|b| Problem 3.8 In this problem, you are asked to prove that for \(a, b \in \mathbf{R}$$ if $$a \leq b_1$$ for every $$b_1 > b$$, then $$a \leq b$$.

I think the easiest way to approach this problem is to do a proof by contradiction. The idea is that in order to prove that a statement is true, you prove derive a contradiction from its negation. The negation of the statement we are trying to prove is, “for all $$b_1 > b$$, $$a \leq b_1$$ and $$a > b$$.”

If you think about what this statement is saying, it should be clear that it cannot possibly be true. For example, if $$a > b$$, then we can choose $$b_1 = (a + b) / 2$$ so that $$b b$$ $$a \leq b_1$$. Since the negation of the statement we are trying to prove leads to a contradiction, the statement must be true.