# A finite set and convergence

1. Nov 3, 2012

### bedi

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

The definition of limit.

3. 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?

2. Nov 3, 2012

### 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 ...

3. Nov 3, 2012

### bedi

Ah okay, thank you

4. Nov 3, 2012

### 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.