Find greatest lower bound and least upper bound

Question

Let $\mathbb{R}$ be the set of real numbers and $≤$ be the partial order on $\mathbb{R}$.
Consider the subset $$ A = \{ x ∈ \mathbb{R}: 1<x<2 \} $$ Find the greatest lower bound and least upper bound of $A$.

Answer 1

$$ A = \{ x ∈ \mathbb{R}: 1<x<2 \} $$ Claim: The least upper bound (l.u.b) of $A$ is $2$, and the greatest lower bound (g.l.b) of $A$ is $1$.

Outline of proof:
To show that $2$ is the l.u.b, show that it is an upper bound for $A$, and that if $u\in\mathbb{R}$ is an upper bound for $A$, then $u\geq 2$.

Similarly to show that $1$ is the g.l.b, show that it is a lower bound for $A$, and that if $l \in\mathbb{R}$ is a lower bound for $A$ then $l\leq 1$.

The above outline really just follows from the definitions, and I hope you can fill in the details.