Proving $\left \{ a_{n}b_{n} \right \}$ diverges to infinity

real-analysissequences-and-series

The question is:

If $\left \{ a_{n} \right \}$ and $\left \{ b_{n} \right \}$ are both sequences that diverge to infinity, prove that the sequence $\left \{ a_{n}b_{n} \right \}$ diverges to infinity.

The problem I am having with this is that, although I know the definition of a divergent sequence, I am not given an explicit sequence. Therefore, I am having trouble formulating an argument.

Can anyone give me a start?

Thanks so much!

Best Answer

Let $A>0$ be given. By definition, there exist $N_1, N_2 \in \mathbb N$ such that \begin{equation}|a_n|>\sqrt A \text{ whenever } n \geq N_1 \end{equation} and \begin{equation}|b_n|>\sqrt A \text{ whenever } n \geq N_2. \end{equation}

Set $N=\max\{N_1, N_2\}$. So we have $|a_n b_n|>A$ whenever $n \geq N$. Therefore, $a_nb_n \to \infty$ as $n \to \infty$.

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