Let sn be a convergent sequence whose range has no accumulat

Let (s_n) be a convergent sequence whose range has no accumulation points. Show that there exists an N such that all terms with n greaterthanorequalto N are equal to one another.

Solution

Condition for the series to be convergent is that it should be bounded

I.e series tends to zero as n tends to infinity.

Given sequence range has no accumulation points which tells that series final values tends to zero after some values of N

Hence for n >= N

All the values of sequence tends to zero

Which in turn are equal to one another.