Convergence of Finite Sets: A Limit on Repeated Elements?

[bedi]

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?

 

[HallsofIvy]

If A= {x_n} is finite then the set of distances {|x_i- x_j|} is finite and so has a smallest value. If $\epsilon$ is smaller than that ...

 

[bedi]

Ah okay, thank you

 

[funcalys]

Idea: Establish a bijection f: N -> A
n |-> f(n)=x_{n}
If there exists no N: $\forall n > N, x_{n} = const$, then A must be infinite -> hence we obtain a contradiction.