Sequence diverging to infinitive. Prove that the sequence diverges to negative infinitive. $ x_n = n - 3n^2$ as $n$ grows. Proof: By definition, a sequence approaches negative infinitive iff for every real number $M$, there exists a natural $N$ such that $n \ge N \to x_n \lt M$. Now, $x_n = n - 3n^2 = n(1 - 3n) \lt 3n$. So $n \ge N$ implies $n - 3n^2 = n(1 - 3n) \lt 3n \lt 3N \lt M$ I am having trouble understanding how to find big $M$. Thank you.

**_Hint:_**

$$ x_n = n - 3n^2 = (n - n^2) -2n^2 $$

You can use induction to prove that $ n - n^2 \le 0 $ whence it follows that $$x_n \le -2n^2 \le -2n $$