[Math] Infimum, supremum of a set problems

Question

I am solving some infimum/supremum problems, and my book has different answers for some of the problems.

Let $A = \{ x \in \Bbb N | x^2 < 5\}$ find sup A and inf A, their answer is sup A = $\sqrt5$, inf A = $-\sqrt5$.

I think this is wrong, since A is a finite set, its clear sup A = 2 and inf A = 0, am i missing something?

Another one, $A = \{x^2+x |x \in (-1, 1)\}$ they say sup A = 1 and inf A = 0, i think sup A = 2 and inf A = 0, again am i wrong?

Answer 1

For the first, I'd say you're right. The official solution woul be for $x \in \mathbb{Q}$ (or $\mathbb{R}$) instead of $x \in \mathbb{N}$.

For the second one, take $x = -\frac{1}{2} \in (-1,1)$ (I assume these are the real numberse between $-1$ and $1$, excluding the borders). Then $$x^2 + x = \frac{1}{4} - \frac{1}{2} = -\frac{1}{4},$$ so $\inf A \le -\frac{1}{4}$.

Hint to solve this:

You want to check the borders of the interval (-1,1) and any minimum/maximum of your expression in that interval, for which derivations will be very helpful. One of the obtained x (should be 3 of them) will give you the infimum.