Answer to Question #3986 in Real Analysis for Junel

Question

let S subset R be nonempty. show that if u= supremum S, then for every number n element N the number (u-1)/n is not an upper bound of S, but the number (u+1)/n is an upper bound of S.

Answer 1

1) Let's prove the first part.
Assume that there is some n element N such that (u-1/n) is an upper bound of S.
By the definition supremum is the lowest upper bound. But easy to see that (u-1/n) < u and thus u isn't the lowest bound.
So it's not true that there is some n element N such that (u-1/n) is an upper bound of S.
It means that for every number n element N the number (u-1/n) is not an upper bound of S.
2) Let's prove the second part.
For any x element S it's true that x <= u, because u = sup S.
At the same time for any n element N it's true that u < u+1/n
So we have for any x element S: x <= u < u+1/n. From the transitivity of real numbers x < u+1/n
Thus for any n element N (u+1/n) is greater than any element of S, which means that (u+1/n) is an upper bound.