-2

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.

DeepSea
  • 77,651
Hy L
  • 15
  • Consider the subset $B = {2x : x \in \mathbb{R}}$. Is $B$ bounded? – Ethan Alwaise Nov 28 '16 at 04:39
  • For this to be a good question, you should explain where your difficulties lie (otherwise it looks like an attempt to outsource homework). Have you problems understanding the set builder notation? What does the set $A$ look like to you? If that is not a problem, then it is just about applying the definition. In this case a comparison with textbook and/or lecture examples will be very illuminating. – Jyrki Lahtonen Nov 28 '16 at 06:18

3 Answers3

4

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.

Retired account
  • 1,080
3

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.

Anthony P
  • 609
1

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

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