Is there a difference in definition between an increasing sequence and a monotonically increasing sequence? Is there a difference in definition between an increasing sequence and a monotonically increasing sequence? In some places I see they call it just inc/decreasing and some call it monotonically inc/decreasing, is there a difference? Can we say that for example, if a sequence increases for only some of its indices then it's increasing but not monotonically? Like for example $(-1)^n n$ which has an increasing subsequence.

A monotone sequence is a sequence that is increasing or decreasing. $$\begin{align} \text{monotone increasing}&=\text{increasing}\\\ \text{monotone decreasing}&=\text{decreasing} \end{align}$$