Need example of a continuous function map cauchy sequence to noncauchy sequenceby xsw001 Tags: cauchy, continuous, function, noncauchy, sequence 

#1
Jan2311, 06:20 PM

P: 37

1. The problem statement, all variables and given/known data
I need a example of a continuous function f:(X, d) > Y(Y, p) does NOT map a Cauchy sequence [xn in X] to a Cauchy sequence of its images [f(xn) in Y] in the complex plane between metric spaces. 2. Relevant equations If a function f is continuous in metric space (X, d), then it continuous at every point in X. Definition of a sequence converges to a point in metric space: If {xn} > x, then for all e>0, there exists an N such that d(x, xn)<e, for all n<= N Definition of Cauchy sequence in metric space:  {xn} is a Cauchy sequence if for all e>0, there exists an N such that d(xn, xm)<e, for all n, m <= N  Every convergent sequence in metric space is a Cauchy sequence. 3. The attempt at a solution I have counterexample in mind in R but not complex plane in metric space though. Let f:R{0} > R{0} defined by f(x)= 1/x, so f(x) is continuous on all domain. Let {xn} = 1/n, then {xn} converges to 0 and therefore is a Cauchy sequence in R–{0}, but its image f(xn) = 1/xn = 1/(1/n) = n diverges, therefore is NOT a Cauchy sequence. But I need an example of a continuous function f:(X, d) > Y(Y, p) does NOT map a Cauchy sequence [xn in X] to the Cauchy sequence of its images [f(xn) in Y] between complex planes in the metric spaces? Any suggestions? 



#2
Jan2311, 06:29 PM

P: 39

1/x?
EDIT: nm i didnt read the whole post 


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