Let f and g be two continuous functions on the closed interval [0,1] such that for every x in [0,1] : f(x)<g(x).
Prove that there exists some number m>0 such that for every x in [0,1] : f(x)+m<g(x).
Since f and g are each continuous, what might be said about the continuity of h = g - f? What must be true of the sign of h on the interval? What does the Extreme Value Theorem say about a minimum value for h on the interval? How might this be useful in your proof?
If you would care for further guidance, kindly please reply with your answers to these questions, along with your thoughts thus far. Thank you.