The question:
Let f(x) and g(x) be continuous on the closed, finite interval [a,b] with f(x)>g(x) for all x∈[a,b]. Prove that there exists an α>0 such that f(x)>α+g(x) for all x∈[a,b].
I tried using ϵ−δ continuity as well as sequential continuity but didn't get anywhere. Any suggestions would be very helpful!
(Note: these are real-valued functions only.)
Answer
Hint: Does α=min satisfy f(x) > g(x) + \alpha for x \in [a,b]?
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