The general term of the sequence
If the geometric sequence with ration is a constant sequence with each term equal to 0.
The term of the sequence is
The sequence is a constant sequence and is bounded
The constant sequence is a always convergent and the sequence is converging to 0.
Therefore,holds
Now it is sufficient to prove the result for
It is observed that
If then
Also,
The sequence is decreasing sequence and is bounded below by 0.
The monitonic decreasing sequence which is bounded below is convergent
Therefore,sequenceis convergent
Assume that
Therefore,
(take limit)
Therefore,the sequenceconverges to 0