Question

Let E be a nonempty subset of an ordered set ; suppose α is a lower bound of E and β is an upper bound of E. Prove that α ≤ β.

Answer 1

Since \(\alpha\) is a lower bound of \(E\), you have \(\alpha\le x\) for all \(x\in E\).

Similarly, since \(\beta\) is an upper bound of \(E\), you have \(\beta\ge x\) for all \(x\in E\). (Or \(x\le \beta\))

So if \(\alpha\le x\) and \(x\le \beta\), what does that tell you?

Answer 2

In case you were wondering, the result follows directly from the definitions of lower/upper bounds.

Answer 3

I know that but why do our teacher give us that us a homework? its so easy we just need to know the definion of lower/upper bounds

Answer 4

Definitions are key to proof-writing. Just because it's simple doesn't mean it's not an important result.

Answer 5

ok thank you :)