Monotone sequence - how can this be a monotone sequence if it bounces between numbers

My understanding of showing a sequence is monotone is to show that it either increases or decreases by working out

$\displaystyle (a_n+1) - (a_n)$

and if it's > 0 its increasing, if less than, decreasing, right?

But now i have this problem that I must say if the sequence is monotone:

$\displaystyle (a_n) = (-1)^n + 2n$

which to me works out that it is bounded by 0 and 4, ie that it bounces between 0 and 4 ( if n is odd then the value is 0 if even then its 4)

Surely this goes against monotone sequence??? But my answer says that it IS monotone. How can it be monotone when it is neither increasing nor decreasing??