Statement :

If converges then .

Theorem 9.5:

If the sequence is bounded and monotonic then it converges.

Thus, is a monotonic sequence, which means strictly increasing or decreasing.

Thus, the difference between consecutive terms should not be zero.

Limit value of such difference is also not equal to zero as tends to zero.

Consider a counter example sequence as .

Find .

The statement is false. 

The statement is false.