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Homework Statement
S≡{x|x∈ℝ,x≥0,x2 < 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