We just had our first calculus lecture, and I'm kinda stuck at this proof right now:
Prove that $\lim_{n\to\infty}\{(-1)^n\}$ diverges.
Given: $\lim_{n\to\infty}(-1)^n = a$. Thus for $\varepsilon_0 = 1 $ there should be $n_0 \in\mathbb N$ so that $$\forall n > n_0; |(-1)^n-a| < 1 $$ But when $n > n_0$, then $$|(-1)^n-(-1)^{n+1}| \le |(-1)^n-a|+|a-(-1)^{n+1}| < 1+1 = 2$$ -> A contradiction!
Now I think I lose it at $$|(-1)^n-(-1)^{n+1}|$$ Why does he substitute 'a' with $(-1)^{n+1}$, or does he anyways?
any help is appreciated !