> In other words, \\(g(b)\\) must be an **[upper bound](https://en.wikipedia.org/wiki/Upper_and_lower_bounds)** of this set. But you shouldn't choose \\(g(b)\\) to be any bigger that it needs to be! After all, you know \\(a \le_A g(b)\\) _only if_ \\(f(a) \le_B b\\). So,
\\(g(b)\\) must be a **[least upper bound](https://en.wikipedia.org/wiki/Infimum_and_supremum)** of the above set.

**Puzzle TR1**. Why *precisely* must \\(g(b)\\) be a least upper bound of the set?

(Maybe this is obvious, but it took me quite a while)