Consider the following statement:(adsbygoogle = window.adsbygoogle || []).push({});

If [tex] \left\{ a_n \right\} [/tex] and [tex] \left\{ b_n \right\} [/tex] are divergent, then [tex] \left\{ a_n b_n \right\} [/tex] is divergent.

I need to decide whether it is true or false, and explain why. The real problem is that I checked the answer in my book; it's false, but I don't understand it. Here is what I think:

Let's suppose that both sequences are convergent. Then, it follows that

[tex] \lim _{n\to \infty} a_n \cdot \lim _{n\to \infty} a_n = \lim _{n\to \infty} \left( a_n b_n \right) \tag{1} [/tex]

But, the truth is that both are divergent. So, [tex] \lim _{n\to \infty} a_n \neq 0 [/tex] and [tex] \lim _{n\to \infty} b_n \neq 0 [/tex]. If neither is zero, then how can [tex] \lim _{n\to \infty} \left( a_n b_n \right) = 0[/tex] (so that the statement is false)? It doesn't sound reasonable if you consider (1).

Can anybody please help me clarify this?

Thank you very much.

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# Homework Help: Let's suppose that both sequences are convergent

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