Answer
False
Work Step by Step
Let us consider an example: $a_n=\dfrac{(-1)^n}{n}$
Now, $a_{n+1}-a_n=\dfrac{(-1)^{n+1}}{n+1}-\dfrac{(-1)^n}{n}=\dfrac{(-1)^{n+1}(2n+1)}{n(n+1)} \ne 0$
This implies that the sequence is not monotonic.
Thus, the given statement is false.