Question

Suppose that\[\lim s_n=0.\]If\[(t_n)\]is a bounded sequence, prove that\[\lim (s_n t_n)=0.\]

Proof:
Let\[(t_n)\]be a bounded sequence. Then \[\left \{ t_n:n\in\mathbb{N} \right \}\]is a bounded set, i.e.,\[\exists M\geq 0\ni \forall n,|t_n|\leq M.\]Let\[\lim s_n=0\]and\[lim t_n=M.\]Then\[\lim (x_n t_n)=0\cdot M=0.\blacklozenge\]

Does this seem plausible?

Answer 1

No, because \( (t_n) \) need not have a limit.

But what you do know is that
\[ |s_nt_n | \leq M |s_n| \]

Answer 2

You're right. I just noticed that\[(-1)^n\]is bounded, but has no limit.

I'm going to see how I can proceed from what you've told me.

Answer 3

JamesJ, this is how I re-wrote it.

Answer 4

You've got the idea, but re-read your proof carefully, as you have some extraneous details in there.