A finite set and convergence

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Homework Statement



Let A be a finite subset of R. For each n in N, let x_n be in A. Show that if the sequence x_n is convergent then it must become a constant sequence after a while.

Homework Equations



The definition of limit.

The Attempt at a Solution



As A is finite, at least one element of A will appear in the sequence more than once after some N. As this sequence is convergent there is an M for any ε such that |x_n - x| < ε for every n with n>M. Let M>N.... Help please.

I can't decide whether this problem is too hard or I'm stupid or this is just because I'm a beginner?
 

Answers and Replies

  • #2
HallsofIvy
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If A= {x_n} is finite then the set of distances {|x_i- x_j|} is finite and so has a smallest value. If [itex]\epsilon[/itex] is smaller than that ...
 
  • #3
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Ah okay, thank you
 
  • #4
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Idea: Establish a bijection f: N -> A
n |-> f(n)=x_{n}
If there exists no N: [itex]\forall n > N, x_{n} = const[/itex], then A must be infinite -> hence we obtain a contradiction. :smile:
 

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