Suppose that a sequence has a finite range. Under what conditions will the sequence converge ?

Answer 1

Suppose the finite set A is the range of a sequence (a_n). If a_n -> a, then a is not a limit point of A, because finite sets have no limit points. It follows there exists a neighborhood V of a that contains at most 1 element of A. Since a_n -> a, for sufficiently large n all a_n are in V, which implies a_n=a. Therefore, (a_n) is eventually constant, there exists a k such that a_n = a for every n>=k.

If (a_n) is eventually constant, then (a_n) is trivially convergent.

So, if (a_n) has a finite range, then (a_n) converges if and only if it is eventually constant.

Answer 2

A sequence with a finite range will converge iff it is eventually constant.

Answer 3

what do you mean with "finite range" ?
do you mean , there are only a fintite number of terms ?, do you mean that each term is finite ( bounded ) .... mysteries