Proof that the sequence $a_{n} = n(-1)^n, n \in \mathbb{N} $ diverges The sequence is $-1,2,-3,4,-5,...$ I understand that it will oscillate between arbitrarily large positive and negative values as $n \rightarrow \infty $. How could I formalize that argument?

Using @Sujaan Kunalan suggestion,

There are subsequences

$a_{2j} = 2j(-1)^{2j} = 2j \rightarrow +\infty$ when $n=2j \rightarrow \infty$

and

$a_{2j+1} = (2j+1) \cdot (-1)^{2j+1} = -(2j+1) \rightarrow -\infty$ when $n=2j+1 \rightarrow \infty$

The original sequence cannot converge and this could be concluded in two ways

1 - There are subsequences which tend to different limits when $n$ tends to $\infty$ (that would be sufficient even if the limits where finite);

2 - There is a subsequence which diverges when $n$ tends to $\infty$