The sequence $\{n(n-1)\}$

Question

I am stuck with the problem on showing that $\{n(n-1)\}$ diverges. I know it leads to the indeterminate form $\infty-\infty$. That is, $$\displaystyle\lim_{n\to\infty}n(n-1)=\displaystyle\lim_{n\to\infty}(n^2-n)=\infty-\infty$$. How do I proceed? Thanks.

Answer 1

For $n> 2$, $n-1 > 1$

Hence we have $n(n-1) > n$

That is $$\lim_{n \to \infty} n(n-1) \ge \lim_{n \to \infty} n = \infty$$

Answer 2

HINT

Let $a_n = n(n-1)$ for $n \ge 1$. Note that $a_{n+1} > a_n$ and in addition, if $M \in \mathbb{R}$ then $a_{M+1} = M(M+1) > M$, so the sequence is unbounded above.