I just asked a question about this kind of stuff so I feel bad asking again, but I could use some help. This is a homework question that reads:
Use the definition of limit to prove that the sequence $(-1)^{(n-1)}$ diverges. Hint: Use the triangle inequality.
I do not really understand how the triangle inequality relates to divergence. I am not necessarily looking for a direct answer to a homework question, but a push in the right direction would be nice. Thanks!
Best Answer
If you must use the triangle inequality, maybe you can do something like this:
Define $a_n=(-1)^{n-1}$ for $n\in\mathbb{N}$. Suppose $(a_n:n\in\mathbb{N})$ is converging with limit $a$. Let $1>\epsilon>0$ then there exists $N>0$ such that for all $n>N$, $|a_n-a|<\epsilon$ but since $$ |a_{n+1}-a|\geq \big||a_{n+1}-a_n|-|a_n-a|\big| > 2 -\epsilon > \epsilon $$ for all $n>N$ this leads to a contradiction from which we conclude that $(a_n)$ diverges.