find a convergent sequence in (0,1] that does not converge to a point in (0,1].

Question

find a convergent sequence in (0,1] that does not converge to a point in (0,1].

Explanation / Answer

a) Since {An} is convergent (call its limit L), for any epsilon > 0, there exists a positive integer N such that |An - L| N. If L = 1 + k for some k>0, then choose epsilon = k. Then, there exists positive integer n such that ==> |An - (1+k)| 1 N. In particular, we have that L > 1. This is a contradiction of An being in [0,1]. Hence, L can't be bigger than 1. Similarly, if L = 1 - k, we get a contradiction (again by setting epsilon = k). Thus, L is in [0,1]. b) Let {An} = {1 - (1/n)} for n>=1. Note that An is in (0,1) for all n>=1, but lim(n--> infinity) An = 1 which is not in (0,1).

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