29.18
Let f be a differentiable on R with a = sup {|f ′(x)|: x in R} < 1.
Select s0 in R and define sn = f (sn-1) for n ≥ 1. Thus s1 = f (s0), s2 = f(s1), etc
Prove that (sn) is a convergencesequence. Hint: To show (sn) is Cauchy, first show that |sn+1 - sn| ≤ aּ|sn - sn-1| for n ≥ 1.
5.2.1. Show that if (a_n)[n=1,infinity] and (b_n)[n=1,infinity] are equivalent sequences of rationals, then (a_n)[n=1,infinity] is a Cauchy sequence if and ...
... depending on ) such that |an - |am < for all n, m >= M. (1) * Show that the sequences ( 1/n) and (n + 1/2n) and Cauchy sequences, but the sequence (n) is not. ...
... is a proof regarding closed subsets and Cauchy sequences. Proof: Let F ⊆ R , then F is a closed set if and only if the limit every convergent sequence in F ...
... are cauchy sequences. Use a triangle inequality argument to prove cn =| an − bn | is cauchy. Definition Definition. A sequence is called a Cauchy sequence if ...
... For the reverse direction, I know I have to take a cauchy sequence and translate it to the unit circle and then show that if it is convergent there, it is ...
... line there is plenty of room to find a converging sequence of different ... the right hand side is complex differentiable (it satisfies the Cauchy-Riemann equations ...
Complex Variable; Uniformly Continous; Lipshitz Function; Cauchy Sequence. ... continuous. Problem #5. First I show that { f ( xn )} is a Cauchy sequence. ...
... exists for each real x (eg by showing that g_k(x) is Cauchy with an ... C_0 such that A is uniformly bounded and equicontinuous, then every sequence of functions ...
