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

1 Answer
Cesareo R.
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,

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