Let $A$ be a subset of $\mathbb{R}$ with the following property: $a \leq 1$ for $a$ in $A$

Question

Let $A$ be a subset of $\mathbb{R}$ with the following property: $a \leq 1$ for $a$ in $A$

1. Give an example that sup$A = 1$

2. Give an example that sup$A < 1$

What does it mean to give examples of such conditions? Does it mean to pick different sets of $A$ that satisfy those conditions?

If so, how does a set have the supremum that $ \leq 1$? Don't all sets have one supremum if exists?

Answer 1

If we take $A = (0, 1)$ this is an example for 1)

If we take $A = (0, 0.5)$ this is an example for 2)

And yes, every subset of $\mathbb{R}$ has a unique supremum (if it exists).

Answer 2

In the way you asked the question I think the aim is to give an example of set $A \subseteq \mathbb{R}$ such all its elements are $\le 1$ and such that $\sup A = 1$ or such that $\sup A < 1$

To answer the first question take for example $A :=(0,1) \subseteq \mathbb{R}$. It clearly has $\sup A = 1$

For the second just take $A:= (-1,0)$.