Monotonic increasing of a sequence

Question

So my question is that if the limit of a sequence goes to infinity, does this sequence have to be increasing?

Answer 1

No it doesn't, it could for example still go up and down while still having a general trend towards infinity - as an example, think about a sequence $(a_n)$ that is equal to $n$ for even $n$ and $n - 2$ for odd $n$. Since we always have $(a_n) \geq n - 2$ this sequence goes to infinity, but it is neither increasing nor decreasing, since for odd $n$ we have $n - 2 = a_n < a_{n+1} = n+1$, but for even $n$ we have $n = a_n > a_{n+1} = n - 1$.

Answer 2

Not of course, let consider

$$a_n= n+2\cos (\pi n)=-1,4,1,6,3,8,5,10,7,...\quad a_n\to \infty$$

Indeed, more in general, any bounded sequence which oscillates (regularly or irregularly) around a value which tends to infinity will satisfy the definition

$$\forall M \quad \exists \bar n: \quad n>\bar n \implies \quad a_n>M$$