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.