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.
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?