The question is: prove that $x_n=n-3n^2$ diverges to $-\infty$ as $n\to\infty$ using only the definition of divergence.
I know that $x_n$ diverges to $-\infty$ iff for each $M\in\mathbb{R}$ there is an $N\in\mathbb{N}$ such that $n\geq N$ implies $x_n<M$ but I don't know how to go about rigorously showing that $x_n$ diverges. Is there a general way of proving divergence?
Best Answer
If $n \geq 1$, then $n-3n^{2} = n(1-3n) \leq -2n$. Let $M < 0$; then $-2n < M$ if $n > |M|/2$; so for all $n \geq \lceil \frac{|M|}{2} \rceil + 1$ we have $n-3n^{2} < M$.