Prove: $\lim\limits_{n\to \infty} a_n ⋅ b_n = \infty$

Question

How do I prove:

Let $\lim\limits_{n\to \infty} a_n = \infty$ and $\lim\limits_{n\to \infty} b_n = \infty$

Prove: $\lim\limits_{n\to \infty} a_n ⋅ b_n = \infty$

Thank you

Answer 1

Notice can find and $\alpha $ such that $0 < \alpha < \lim b_n$. For large $n$, in particular, $b_n > \alpha. $(proof?)

Let $M > 0$ . Therefore, by hypothesis we can find $N$ such that $a_n > \frac{M}{\alpha} $ for all $n > N $

Therefore,

$$ a_nb_n > \alpha \frac{M}{ \alpha} > M $$

$$ \therefore (a_nb_n) \to \infty $$

Answer 2

Hint: For $n$ large enough, $b_n \geq 1$ and $a_n >0$ hence $a_n \cdot b_n \geq a_n$.