1. The problem statement, all variables and given/known data
Let f : (a, b) → R be a continuous function on (a, b) such that |f'(x)| <= 1 for all x that are elements of (a,b). Prove that f is uniformly continuous function on (a,b).
2. Relevant equations
3. The attempt at a solution
Proof:For the sequence {xn}, where the limit(n→∞) xn = xo, then limit(n→∞) |((f(xn)-f(xo))/(xn-xo))| <= 1. This can be rewritten as limit(n→∞) |(f(xn)-f(xo)) <= limit(n→∞) |xn-xo|. For uniformly continuous, for all ε>0 and x,y that are elements of (a,b), there exists a δ such that |x-y|<δ implies |f(x)-f(y)|<ε. Let x=Xn and y=Xo. Then if ε=δ, then limit(n→∞) |(f(xn)-f(xo)) <= limit(n→∞) |xn-xo|<δ=ε and therefore f is uniformly continuous on (a,b)
Any help on where to improve this or if there are any steps that are incorrect would be greatly appreciated.
Let f : (a, b) → R be a continuous function on (a, b) such that |f'(x)| <= 1 for all x that are elements of (a,b). Prove that f is uniformly continuous function on (a,b).
2. Relevant equations
3. The attempt at a solution
Proof:For the sequence {xn}, where the limit(n→∞) xn = xo, then limit(n→∞) |((f(xn)-f(xo))/(xn-xo))| <= 1. This can be rewritten as limit(n→∞) |(f(xn)-f(xo)) <= limit(n→∞) |xn-xo|. For uniformly continuous, for all ε>0 and x,y that are elements of (a,b), there exists a δ such that |x-y|<δ implies |f(x)-f(y)|<ε. Let x=Xn and y=Xo. Then if ε=δ, then limit(n→∞) |(f(xn)-f(xo)) <= limit(n→∞) |xn-xo|<δ=ε and therefore f is uniformly continuous on (a,b)
Any help on where to improve this or if there are any steps that are incorrect would be greatly appreciated.
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