Question

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

Answer 1

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,