How do you prove that a sequence diverges?

Question

Specifically the sequence $\{(-2)^n\}$

Answer 1

Suppose there exists L such that $(-2)^n$ tends to L. Therefore we know that there exists a positive integer $N$ such that $|(-2)^N-L|<0.5$ and $|(-2)^{N+1}-L|<0.5$ Using the triangle inequality we have: $|(-2)^N-(-2)^{N+1}|<=|(-2)^N-L|+|(-2)^{N+1}-L|<1$ , thus $|(-2)^N||1--2|<1$ (a contradiction)

Answer 2

You have one subsequence decreasing to $-\infty$ and another increasing exponentially to $\infty$. No convergent sequence can have two distinct subsequential limits.