Find the supremum and infimum of $\left \{ n \in \mathbb{N}: n^{2} < 10 \right \}$
Sup $= 3$ makes sense because if $n = 3$, then $n^{2} = 9 < 10$. $n = 4$ is also an upper bound but $3 < 4$ so sup $= 3$ still stands.
Now the answer says that inf $= 1$. I can see how that's a lower bound for the set, but how is it the greatest lower bound? Isn't $1$ the smallest natural number already? What is smaller than it?
Best Answer
Observe that $3$ is an upper bound of $S = \{n \in\mathbb{N} \space \colon n^2 < 10\} = \{1,\space 2, \space 3\}$. Thus $\text{sup(S)} \leq 3$, but $3 \in S$, so $ 3\leq \text{sup(S)}$. Thus $\text{sup(S)} = 3$. Similarly you can show that $\text{inf(S)} = 1$.