If f:[0,1]->RR,|f'(x)|<1 how do you show f(1/n) converges ?

1 Answer
Oct 19, 2017

See below.

Explanation:

Assuming that f(x) in CC^2 we have by Rolle's theorem that there exists xi in (0,1) such that

f(1/n)-f(1/(n+1)) = f'(xi)(1/(n+1)-1/n) and then

abs(f(1/n)-f(1/(n+1))) = abs(f'(xi))(1/(n(n+1)))

but abs(f'(xi)) < 1 so

abs(f(1/n)-f(1/(n+1))) lt 1/(n(n+1)) and them f(1/n) converges according to Cauchy's criterion,