Assistance with phrasing and understanding of a limits/sequences problem.

Mar 2019
1
0
Australia
The following question has been puzzling me for a few days:

Assume (an)n=1 is a convergent sequence of integers. Prove the existence of N ∈ N such that ai = aj for all i, j > N.

I am failing to interpret the meaning of the ai = aj . I understand that I need to prove the existence of a cutoff point, meaning I will probably need to fix an epsilon, etc. Using triangle inequalities may also be useful, I'm not sure, maybe none of those ideas are right.

I am still fairly new to calculus so any help understanding this problem and would be appreciated.

K.
 

romsek

MHF Helper
Nov 2013
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California
$a_i, ~a_j,~\text{are just two elements of the integer sequence}$

It's saying that if this sequence of integers converges that if you go far enough along the sequence all elements become equal.

This must be so as non-equal integers are always at least 1 apart.
 
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Plato

MHF Helper
Aug 2006
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Assume (an)n=1 is a convergent sequence of integers. Prove the existence of N ∈ N such that ai = aj for all i, j > N.
There is a well known theorem: Every convergent sequence has the property that from some point on all terms are (infinitesimally) close together. Well that is impossible for integers unless the sequence of integers unless that sequence becomes constant.
 
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