Let x be an upper bound for S subset of R(real). If x is an element of S, prove

Question

Let x be an upper bound for S subset of R(real). If x is an element of S, prove that x=supS. Please show all steps. Thanks!

Explanation / Answer

First assume x = sup S. By the denition of the supremum of a set, x is an upper bound of S so (a) is satised. To prove that (b) is satised, we will use a proof by contradiction. That is, assume there exists some > 0 such that for every s 2 S, x ?? s. But this implies x ?? is an upper bound for S. Moreover, x ?? 0, and this contradicts the denition of x as the least upper bound of S. Conversely, assume x 2 R satises both (a) and (b). Showing x = sup S is equiv- alent to showing x is an upper bound for S, and if x0 is any other upper bound, then x x0. By property (a), we can immediately conclude x is an upper bound for S. Now suppose there exists an upper bound x0 of S with x0 0 so taking = x??x0 and applying property (b) shows the existence and an s 2 S such that s > x ?? () = x ?? (x ?? x0) = x0. This contradicts the fact that x0 was an upper bound for S. Hence x x0 and x = sup S.

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