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## Homework Statement

S≡{x|x∈ℝ,x≥0,x

^{2}< c} Prove Sup S = c

## Homework Equations

## The Attempt at a Solution

Since x in the set real numbers, there are two cases for x: x < 1 or 0 <= x <=1

if 0 <= x <=1, then x < c + 1 since c is positive.

if x < 1, then x^2 < c < x*c + x = x(c+1)

thus x < c+1, by the completeness axiom, S has a least upper bound, let's called it b.

I am stuck at this step, can anyone give me a hit or a suggestion to keep going. Thanks