- #1
DivGradCurl
- 372
- 0
Consider the following statement:
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
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.
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.