Infimum and supremum of set $A$

Question

Find the infimum and supremum of the set $A = \{x+y: x, y \in \mathbb{R} \}$ and $x,y$ are real numbers or prove that they do not exist.

Answer 1

Hint: $\mathbb{R}=\mathbb{R}+\mathbb{R}$

So $A=\mathbb{R}$ and thus its infimum is $-\infty$ and its supremum is $+\infty$.

Answer 2

Suppose there exists a supremum for the set such at $\sup(A) = n$. Then $n \geq (x+y) \quad \forall x,y \in \mathbb{R}$. The only such $n$ would be $\infty$, the same argument being that the infimum would have to extend to $-\infty$, neither of which are valid as the sup or inf need be finite.

Answer 3

Notice that with $y=0$, your set is actually just R, so R is a subset of this set. But R is also closed under addition, so this set is a subset of R. Hence it is R, and R is unbounded, it does not have sup or inf(without any bounds none can be the greatest or least).

Hence, your set does not posses sup or inf.