... such that x = p Proof: Let S = {r ∈ Suppose that x < : r > and r < p} and x = sup S • x • ( x+ ) 2 n p • p We want to find something positive, say 1/n, that we can add on to x and still have its ... above by 8, 9, 8.5, π , and any other real number greater than or equal to 8.Since ∈ S, we have max S = Similarly, S has many lower bounds, including 2, which is the largest of the lower bounds and ... – is not an upper bound of = m such that n > m – this contradicts m being an upper bound of But then n +and > m, since n + ∈ , ♦ The Archimedean Property is widely used in analysis and there...