1
$\begingroup$

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

$\endgroup$
1
  • 1
    $\begingroup$ do you at least feel that this should be true somehow? $\endgroup$
    – user87543
    Nov 1 '13 at 10:37
3
$\begingroup$

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

$\endgroup$
0
$\begingroup$

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,

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

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.